CN110687781A - Accurate uncertainty and control gain estimation method of second-order nonlinear system - Google Patents

Accurate uncertainty and control gain estimation method of second-order nonlinear system Download PDF

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CN110687781A
CN110687781A CN201911058957.0A CN201911058957A CN110687781A CN 110687781 A CN110687781 A CN 110687781A CN 201911058957 A CN201911058957 A CN 201911058957A CN 110687781 A CN110687781 A CN 110687781A
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nonlinear system
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刘陆
岳佳旺
古楠
李永明
王丹
彭周华
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Dalian Maritime University
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    • 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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Abstract

The invention discloses a method for estimating the accurate uncertainty and the control gain of a second-order nonlinear system, which estimates by using the accurate uncertainty and the control gain estimation structure of the second-order nonlinear system, wherein the accurate uncertainty and the control gain estimation structure of the second-order nonlinear system comprises an extended state observer and a control gain estimation module. The invention establishes the extended state observer and the control gain estimation module, realizes the simultaneous online estimation of the uncertainty and the control gain of the nonlinear system, and provides convenience for the design process of the controller of the nonlinear system. The invention establishes the control gain estimation module, can realize the on-line estimation of the control gain of the nonlinear system in limited time, and effectively ensures the real-time performance and the accuracy of the parameter estimation. The invention only carries out online estimation of unknown items based on the input and output information of the nonlinear system, and is used for estimating the uncertainty and the control gain of the high-order nonlinear system.

Description

Accurate uncertainty and control gain estimation method of second-order nonlinear system
Technical Field
The invention relates to the field of nonlinear systems, in particular to a method for estimating the accurate uncertainty and control gain of a second-order nonlinear system.
Background
Since the 21 st century, the modern control theory has obviously advanced in all aspects, the technology thereof has penetrated into the subjects of filtering technology, artificial intelligence technology, system engineering and the like, and the linear system control theory has been widely paid attention as an important research direction. However, linear system control is often only suitable for simple phenomena such as attenuation, free oscillation, and infinite divergence, and in the field of actual engineering techniques, complex and various phenomena such as self-sustained oscillation, multiple equilibrium states, chaos, etc. may occur, and at this time, the linear system cannot describe some important nonlinear characteristics of the systems, and the nonlinear system theory solves the analysis and control problems of most models under realistic conditions. In recent years, particularly with the explosive development of artificial intelligence technology and intelligent control technology, nonlinear system theory has become a popular research field, in which method research for estimating unknown terms of nonlinear systems has attracted extensive attention.
For a nonlinear system, the uncertainty of the system and the control gain are often unknown. Originally, people used a PID controller parameter estimation method based on error control, but this method often had the shortcomings of the controlled object output being not jump, differential signal not easy to extract, closed loop system reaction being slow, dynamic characteristics being poor under no disturbance, etc. Later, people establish an extended state observer based on errors and disturbances for parameter estimation, wherein the principle of the extended state observer is to expand the disturbance effect affecting the controlled input into a new state variable, and establish a special feedback mechanism to observe the expanded state, so that the extended state does not depend on a specific mathematical model for generating the disturbances and does not need to directly measure the effect of the disturbance. However, the existing method based on the extended state observer still has the following problems:
first, in the existing estimation method of the adaptive nonlinear system based on the extended state observer, only the uncertainty of the system can be estimated, and the control input gain is not simultaneously estimated on line. The control input gain is used as an important gain parameter and can intuitively reflect the size of the control input, so that the uncertainty and the control gain of the nonlinear system can be estimated simultaneously, and the control characteristics of the whole nonlinear system can be more comprehensively embodied.
Secondly, in the existing estimation method of the adaptive nonlinear system based on the extended state observer, the estimated value can not be guaranteed to be converged to the true value. Meanwhile, the system can not adapt to the change of parameters within a limited time, and the real-time performance and the accuracy of the nonlinear system can not be effectively ensured.
Thirdly, in the existing estimation method of the adaptive nonlinear system based on the extended state observer, the estimation of the control gain of the high-order nonlinear system is not realized, the state that the first-order system can represent is limited, and when a complex control system in multiple states is encountered, the first-order system cannot represent the system.
Disclosure of Invention
In order to solve the defects of the prior art, the invention designs an accurate uncertainty and control gain estimation method of a second-order nonlinear system, which can not only estimate the state function of the system, but also carry out online estimation on control input gain, and ensure gradual estimation in limited time to adapt to the change of parameters.
In order to achieve the purpose, the technical scheme of the invention is as follows: a method for estimating the accurate uncertainty and the control gain of a second-order nonlinear system comprises the steps of estimating by using an accurate uncertainty and control gain estimation structure of the second-order nonlinear system, wherein the accurate uncertainty and control gain estimation structure of the second-order nonlinear system comprises an extended state observer and a control gain estimation module, and the input end of the extended state observer is respectively connected with the output ends of the second-order nonlinear system and the control gain estimation module and an external control input; the control gain estimation module comprises a high-pass filter and an integral filtering regression equation, wherein the input end of the high-pass filter is respectively connected with the output ends of the second-order nonlinear system and the extended state observer and the external control input, and the input end of the integral filtering regression equation is connected with the output end of the high-pass filter; the input end of the second-order nonlinear system is connected with an external control input;
the estimation method comprises the following steps:
A. establishing a second-order nonlinear system
The second order nonlinear system is described by the following differential equation:
wherein x is1,x2Representing the state of a second-order nonlinear system, u representing an external control input, b0And f (-) represents the unknown uncertainty of the second-order nonlinear system.
B. Establishing extended state observer
The dilated state observer is described by the following differential equation:
Figure BDA0002257342290000031
wherein the content of the first and second substances,representing a second order nonlinear system state x1,x2Is detected by the measured values of (a) and (b),
Figure BDA0002257342290000033
an observed value, k, representing an output signal of the uncertainty estimation unit1,k2,k3Representing the known state gain.
C. Establishing control gain estimation module
The control gain estimation module comprises a high-pass filter and an integral filtering regression equation
C1, establishing the following time domain differential equation:
Figure BDA0002257342290000034
wherein the content of the first and second substances,
Figure BDA0002257342290000035
representing unmanned ship state x1Second derivative of (2)
Inputting the above formula into a high-pass filter, and performing laplace transform to obtain:
Figure BDA0002257342290000036
where s denotes a complex variable in the frequency domain, x1And(s) u(s) respectively represent the unmanned ship state and the control input in the frequency domain state, and D(s) is a representation form after Laplace transform of a high-pass filter.
C2, establishing integral filtering regression equation
Order:
Figure BDA0002257342290000037
Figure BDA0002257342290000038
then equation (4) is:
g(s)＝N(s)b0(7)
where g(s) represents the state derivative of the high-pass filter and N(s) represents the filtered regression matrix. Equation (7) is transformed into a time domain equation:
g(t)＝N(t)b0(8)
where g (t) represents the state derivative of the high-pass filter in the time domain, and N (t) represents the regression matrix after filtering in the time domain.
The following integral filtering regression equation is established:
m is the regression after integral filtering, and G is an integral filtering variable. Further expressions for G derived from formulas (8) - (9) are:
G＝Mb0(10)
establishing a storage stackj＝1,2,...,p。
Wherein (N)j,gj) Data indicating the time from j to 1 is respectively
Figure BDA0002257342290000043
The time is stored in the stack W, p ∈ N+To be the length of the stack,
Figure BDA0002257342290000044
using the data stored in stack W, equation (8) is transformed into the following matrix form:
[g1,g2,…gp]＝[N1,N2,…Np]b0(11)
at tcBefore time, parameter b0The estimation method of (2) is a self-adaptive estimation method; when the time reaches tcAfter the moment, the data stored in stack W reaches a maximum value, and the least squares solution of equation (10) is found as follows:
b0＝(MTM)-1MTG
establishing a parameter b0The online estimation equation of (1):
Figure BDA0002257342290000045
wherein
Figure BDA0002257342290000046
Is a parameter b0The on-line estimation of (a) is performed,
Figure BDA0002257342290000047
is a scalar gain used to adjust the convergence rate.
Compared with the prior art, the invention has the following beneficial effects:
first, compared with the existing gain estimation method of the adaptive nonlinear system, the method of the invention establishes the extended state observer and the control gain estimation module, thereby realizing the simultaneous online estimation of the uncertainty and the control gain of the nonlinear system, and providing convenience for the design process of the controller of the nonlinear system.
Secondly, compared with the existing gain estimation method of the self-adaptive nonlinear system, the method has the advantages that the control gain estimation module is established, so that the control gain of the nonlinear system can be estimated on line within a limited time, and the real-time performance and the accuracy of parameter estimation are effectively ensured.
Thirdly, compared with the existing gain estimation method of the self-adaptive nonlinear system, the method disclosed by the invention can be used for carrying out unknown online estimation only on the basis of the input information and the output information of the nonlinear system, and can be expanded to a high-order nonlinear system for estimating the uncertainty and controlling the gain of the high-order nonlinear system.
Drawings
The invention is shown in the attached figure 5, wherein:
FIG. 1 is a schematic diagram of the method of accurate uncertainty and control gain estimation for a second order nonlinear system.
FIG. 2 is a second order nonlinear system control state x1And (5) observing an effect graph.
FIG. 3 is a second order nonlinear system control state x2And (5) observing an effect graph.
Fig. 4 is a graph of the output signal σ observation effect of the second-order nonlinear system uncertainty estimation unit.
FIG. 5 shows the control gain b of a second order nonlinear system0And (5) observing an effect graph.
Detailed Description
The invention will be further described with reference to the accompanying drawings. The structure and flow diagram of the precise uncertainty and control gain estimation method of the second-order nonlinear system related by the invention are shown in fig. 1. Outputting system control state x by external control input u through second-order nonlinear system1、x2To establish an expanded shapeA state observer for inputting external control input u and system control state x1、x2And an observed value of a control gain of the system
Figure BDA0002257342290000051
Obtaining observed values of output signals of a second-order nonlinear system uncertainty estimation unit
Figure BDA0002257342290000052
Then, the observed value of the output signal of the second-order nonlinear system uncertainty estimation unit is estimated
Figure BDA0002257342290000053
System control state x1And introducing an external control input u into a high-pass filter to obtain a state derivative g of the filter and a filtered regression matrix N, and inputting the state derivative g and the filtered regression matrix N into an integral filtering regression equation to obtain a control gain observed value of a second-order nonlinear system
Figure BDA0002257342290000054
And then the data is introduced into an extended state observer to achieve convergence. The invention aims to enable a second-order nonlinear system to realize uncertainty f (-) and control gain b of the second-order nonlinear system under the condition of satisfying the formulas (2) - (12)0Accurate estimation of.
The simulation results are shown in fig. 2-4. FIG. 2 shows a second order nonlinear system control state x1Observation of the Effect, FIG. 3 shows the second order nonlinear System control State x2Observation effect, fig. 4 shows the observation effect of the output signal σ of the uncertainty estimation unit of the second-order nonlinear system, and fig. 5 shows the control gain b of the second-order nonlinear system0And (5) observing the effect. As can be seen from the simulation result diagram, the observed parameters and the actual parameters are converged, that is, the uncertainty f (-) and the control gain b of the second-order nonlinear system are corrected by the method0Accurate estimation is achieved.
The present invention is not limited to the embodiment, and any equivalent idea or change within the technical scope of the present invention is to be regarded as the protection scope of the present invention.

Claims (1)

1. A method for estimating the accurate uncertainty and control gain of a second-order nonlinear system is characterized by comprising the following steps: estimating by using an accurate uncertainty and control gain estimation structure of a second-order nonlinear system, wherein the accurate uncertainty and control gain estimation structure of the second-order nonlinear system comprises an extended state observer and a control gain estimation module, and the input end of the extended state observer is connected with the output ends of the second-order nonlinear system and the control gain estimation module and an external control input respectively; the control gain estimation module comprises a high-pass filter and an integral filtering regression equation, wherein the input end of the high-pass filter is respectively connected with the output ends of the second-order nonlinear system and the extended state observer and the external control input, and the input end of the integral filtering regression equation is connected with the output end of the high-pass filter; the input end of the second-order nonlinear system is connected with an external control input;
the estimation method comprises the following steps:
A. establishing a second-order nonlinear system
The second order nonlinear system is described by the following differential equation:
Figure FDA0002257342280000011
wherein x is1,x2Representing the state of a second-order nonlinear system, u representing an external control input, b0Representing the control gain of the second-order nonlinear system to be measured, and f (-) representing the unknown uncertainty of the second-order nonlinear system;
B. establishing extended state observer
The dilated state observer is described by the following differential equation:
Figure FDA0002257342280000012
wherein the content of the first and second substances,representing a second order nonlinear system state x1,x2Is detected by the measured values of (a) and (b),
Figure FDA0002257342280000014
an observed value, k, representing an output signal of the uncertainty estimation unit1,k2,k3Representing the known state gain;
C. establishing control gain estimation module
The control gain estimation module comprises a high-pass filter and an integral filtering regression equation
C1, establishing the following time domain differential equation:
Figure FDA0002257342280000015
wherein the content of the first and second substances,
Figure FDA0002257342280000016
representing unmanned ship state x1Second derivative of (2)
Inputting the above formula into a high-pass filter, and performing laplace transform to obtain:
Figure FDA0002257342280000021
where s denotes a complex variable in the frequency domain, x1(s) u(s) respectively represent the unmanned ship state and the control input in the frequency domain state, and D(s) is a representation form after Laplace transform of a high-pass filter;
c2, establishing integral filtering regression equation
Order:
Figure FDA0002257342280000022
Figure FDA0002257342280000023
then equation (4) is:
g(s)＝N(s)b0(7)
wherein g(s) represents the state derivative of the high-pass filter, and N(s) represents the filtered regression matrix; equation (7) is transformed into a time domain equation:
g(t)＝N(t)b0(8)
wherein g (t) represents the state derivative of the high-pass filter in the time domain, and N (t) represents the regression matrix after filtering in the time domain;
the following integral filtering regression equation is established:
Figure FDA0002257342280000024
m is a regression quantity after integral filtering, and G is an integral filtering variable; further expressions for G derived from formulas (8) - (9) are:
G＝Mb0(10)
establishing a storage stack
Figure FDA0002257342280000025
Wherein (N)j,gj) Data indicating the time from j to 1 is respectively
Figure FDA0002257342280000026
The time is stored in the stack W, p ∈ N+To be the length of the stack,
Figure FDA0002257342280000027
using the data stored in stack W, equation (8) is transformed into the following matrix form:
[g1,g2,…gp]＝[N1,N2,…Np]b0(11)
at tcBefore time, parameter b0The estimation method of (2) is a self-adaptive estimation method; when the time reaches tcAfter the moment, the data stored in the stack W reaches the maximum value, and the maximum value is obtainedThe least squares solution of equation (10) is as follows:
b0＝(MTM)-1MTG
establishing a parameter b0The online estimation equation of (1):
wherein
Figure FDA0002257342280000032
Is a parameter b0The on-line estimation of (a) is performed,
Figure FDA0002257342280000033
is a scalar gain used to adjust the convergence rate.
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