CN107168071B - A kind of nonlinear system Auto-disturbance-rejection Control based on interference observer - Google Patents
A kind of nonlinear system Auto-disturbance-rejection Control based on interference observer Download PDFInfo
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Abstract
The present invention relates to a kind of nonlinear system Auto-disturbance-rejection Control based on interference observer, for the non-linear single-input single-output system containing unknown nonlinear function and harmonic wave interference；First, harmonic wave interference and unknown nonlinear function are subjected to mathematical character, establish the external model of harmonic wave interference；Secondly, based on control input and measurement output information design interference observer and extended state observer, the estimation to harmonic wave interference, unknown nonlinear function and system mode is completed；Finally, automatic disturbance rejection controller is designed according to the output of interference observer and extended state observer, theoretical based on separation theorem and POLE PLACEMENT USING, the gain for completing observer and controller solves；The present invention has the advantages that strong antijamming capability, control accuracy are high, available for the high-precision control containing interference and Unknown Nonlinear Systems.
Description
Technical field
The present invention relates to a kind of nonlinear system Auto-disturbance-rejection Control based on interference observer, it is possible to achieve harmonic wave is done
Disturb with estimation while unknown nonlinear function with offsetting, controlled available for the system containing harmonic wave interference and unknown nonlinear function
In.
Background technology
Complex control system, especially nonlinear system, it is a variety of not to suffer from model uncertainty, unknown nonlinear etc.
Certainty dynamically influences, and seriously constrains the lifting of control performance.Including modern control theories such as optimum control, robust controls
Control strategy inside is all based on the ideal model design of system, often only considers a kind of or a kind of interference, have ignored very
More unknown non-linear factor, conservative are larger.For example, LQG control considerations is exactly to be by what single white Gaussian noise was influenceed
System, and when system has unknown nonlinear dynamic, its performance can drastically decline.Due to unknown nonlinear dynamic and system shape
State, control input and model parameter etc. intercouple, and cause the information such as its amplitude, frequency to be difficult to determine, therefore, control system
The middle dynamic compensation problem of unknown nonlinear is always one of difficult point of control theory.Result from the PID of last century the '20s
Control due to its is simple in construction, independent of system model the advantages that so that it is in Industry Control always so far
Ascendancy.However, PID control also has its limitation：First, PID control have ignored the information of system model completely；Secondly,
Differential signal in PID control is often difficult to preferably obtain, and easily produces high-frequency noise；Again, the phase that integral element is brought
After steric retardation and vibration etc. consequence；Finally, the tune reference of PID control is cumbersome.For above-mentioned limitation, from PID control,
Han Jing Puritanisms award Active Disturbance Rejection Control (ADRC) method for proposing and possessing disturbance compensation ability, include Nonlinear Tracking Differentiator, expansion shape
State observer and nonlinear feedback controller three parts are formed, and are possessed actively and whole estimation and the ability of compensating disturbance, into
Work(is applied in linear and nonlinear system.ADRC is realized dynamically real-time to unknown nonlinear by extended state observer
Estimation and compensation, effectively improve the robustness and accuracy of complex control system, overcome modern control theory and too rely on
In the limitation of system model.For example, patent authorization number is ZL200410070983.2, Application No. 201510359468.4
Auto-disturbance-rejection Control is employed in patent, non-linear is compensated all disturbances and as total disturbance.
In addition to unknown nonlinear dynamic, many real systems usually contain the harmonic wave interference of the multi-sources such as vibration, and are permitted
The frequency of multi resonant wave interference often can with it is previously known or pass through measure determine.Active disturbance rejection is for the harmonic wave interference of given frequency
Often effect is unsatisfactory for estimated capacity, mainly there is both sides reason：One, traditional Active Disturbance Rejection Control has not utilized interference
Know information, be simply equivalent to the disturbance of rate of change bounded, conservative is larger；Two, if Active Disturbance Rejection Control will lift harmonic wave
For the estimation performance of interference, it is necessary to increase the bandwidth of extended state observer, this often brings larger overshoot and amplification to make an uproar
Sound etc. adversely affects.For such limitation, the control (DOBC) based on interference observer can be with the basis of interference models
The accurate estimation and compensation of frequency information known disturbances are completed, takes full advantage of the characteristic information of interference.For example, patent authorization number
It is humorous during DOCBC realizes filtering or controlled to be employed in ZL200910086897.3, ZL200910086896.9 patent
The estimation and compensation of wave interference.Due to interference with unknown nonlinear dynamic be mutually mixed with coupling, and Interference Estimation error and
Nonlinear Dynamic evaluated error is influenced each other, and accurate control of the DOBC methods also without system in the presence of solving both simultaneously is asked
Topic.The research effectively combined on DOBC and ADRC is not yet found at present.Therefore, for containing harmonic wave interference and unknown non-simultaneously
The nonlinear system of linear function completes complication system harmonic wave, non-linear, it is necessary to make full use of ADRC and the respective advantages of DOBC
A variety of probabilistic counteractings such as dynamic and suppression, so as to realize the high-precision control performance of system.
The content of the invention
The technology of the present invention solves problem：System model, control accuracy are too dependent on for existing control method
It is low, and the problem of be difficult to harmonic wave interference and unknown nonlinear function while compensate, there is provided one kind possesses harmonic wave interference
And Active Disturbance Rejection Control based on interference observer of the unknown nonlinear function estimation in real time with offsetting ability, the present invention has anti-dry
The advantages that ability is strong, control accuracy is high is disturbed, available for the high-precision control containing interference and Unknown Nonlinear Systems.
The present invention technical solution be：A kind of nonlinear system Auto-disturbance-rejection Control based on interference observer,
For the nonlinear system containing harmonic wave interference Yu unknown nonlinear function, first, by harmonic wave interference and unknown nonlinear letter
Number carries out mathematical character, establishes the external model of harmonic wave interference；Secondly, based on control input and measurement output information design interference
Observer and extended state observer, complete the estimation to harmonic wave interference, unknown nonlinear function and system mode；Most
Afterwards, automatic disturbance rejection controller is designed according to the output of interference observer and extended state observer, based on separation theorem and limit
Allocation Theory, the gain for completing observer and controller solve；Specific implementation step is as follows：
(1) harmonic wave interference and unknown nonlinear function are subjected to mathematical character, establish the external model of harmonic wave interference：
Consider the single-input single-output nonlinear system containing harmonic wave interference and unknown nonlinear function as follows：
Wherein, n>0 is the order of system, and y represents model output, y(n)Y n order derivatives are represented,Represent the one of y
Order derivative；f(y,y(1),...,y(n-1)) the unknown nonlinear function that single order can be led is represented, represent Nonlinear Dynamic；B is known
Constant, u are control input, ymRepresent to measure output；Harmonic wave interference known to disturbing d expression frequency informations, can be characterized asWherein, D withUnknown amplitude and phase are represented respectively, and ω is given frequency, and t represents the moment；
Harmonic wave interference d can be described by following external model：
Wherein, w be external model state, coefficient matrixV=[10]；
Unknown nonlinear functionMeet single order can sliver part, i.e.,Its
In, h is unknown bounded function.
(2) based on control input and measurement output information design interference observer and extended state observer, completion pair
The estimation of harmonic wave interference, unknown nonlinear function and system mode；
It is for harmonic wave d design interference observers：
Wherein,W estimate is represented,RepresentFirst derivative,For y estimate,For estimating for harmonic wave interference d
Evaluation, L are the gain matrix of interference observer；
In order to eliminate the influence of unknown nonlinear function, it is defined as follows：
Then nonlinear system can be expressed as the form of augmented system：
Wherein, x1,x2,...,xn+1For the state of augmented system；
With reference to estimation of the interference observer to harmonic wave interferenceExtended state observer is designed to estimate unknown nonlinear letter
Number and system mode：
Wherein,State x is represented respectively1,x2,...,xn+1Estimate；For extended state observer
Output, represent y estimate；k1,k2,...,kn+1For observer gain；
The error for defining interference observer and extended state observer is respectivelyI=1,
2 ..., n+1 can then obtain error dynamics and be：
Wherein, the rank matrix of (n+1) × 1N+1 rank square formations(n+1) × 2 rank square
Battle array(n+1) × 1 rank matrix1 × (n+1) rank Matrix C=[1 0 ... 0], 0 is the zero moment of appropriate dimension
Battle array.
(3) automatic disturbance rejection controller is designed according to the output of interference observer and extended state observer, it is fixed based on separation
Reason is theoretical with POLE PLACEMENT USING, and the gain for completing observer and controller solves；
Output based on interference observer and extended state observer designs the automatic disturbance rejection controller of following form：
Wherein, symbol ∑ represents summation operation, p1,p2,...,pnRepresent controller gain；
System dynamics equation is carried it into, can be obtained：
Based on linear system separation theorem, interference observer gain matrix L and extended state observer gain matrix K=
[k1 k2 … kn+1]TAnd controller gain p1,p2,...,pnIt can be solved respectively by POLE PLACEMENT USING：
Wherein, s represents complex variable, and I represents the unit matrix of appropriate dimension, symbol | | represent to solve the ranks of square formation
Formula, ω0>0、ω1>0 is given constant, represents the bandwidth of system.
The present invention compared with prior art the advantages of be：The present invention is by interference observer and extended state observer phase
With reference to mode complete the On-line Estimation and compensation problem of harmonic wave interference and unknown nonlinear function, take full advantage of interference
Characteristic information, harmonic wave interference and estimation and compensation problem while unknown nonlinear function are realized, overcomes single active disturbance rejection control
Device processed is difficult to compensate for the limitation of harmonic wave interference, available for the high-precision control containing interference and Unknown Nonlinear Systems.
Brief description of the drawings
Fig. 1 is a kind of FB(flow block) of the nonlinear system Auto-disturbance-rejection Control based on interference observer.
Embodiment
Below in conjunction with the accompanying drawings and embodiment the present invention is described in more detail.
As shown in figure 1, present invention specific implementation step is as follows：
The first step, harmonic wave interference and unknown nonlinear function are subjected to mathematical character, establish the outside mould of harmonic wave interference
Type：
Consider the single-input single-output nonlinear system containing harmonic wave interference and unknown nonlinear function as follows：
Wherein, n>0 is the order of system, and y represents model output, y(n)Y n order derivatives are represented,Represent the one of y
Order derivative；f(y,y(1),...,y(n-1)) the unknown nonlinear function that single order can be led is represented, represent Nonlinear Dynamic；B is known
Constant, u are control input, ymRepresent to measure output；Harmonic wave interference known to disturbing d expression frequency informations, can be characterized asWherein, D withUnknown amplitude and phase are represented respectively, and ω is given frequency, and t represents the moment；At this
In inventive embodiments, second-order system is considered, i.e. n=2 takes the unknown nonlinear function to beTake b
=1, D=0.05,ω=20；
Harmonic wave interference d can be described by following external model：
Wherein, w be external model state, coefficient matrixV=[1 0]；
Unknown nonlinear functionMeet single order can sliver part, i.e.,Its
In, h is unknown bounded function.
Second step, based on control input and output information design interference observer and extended state observer are measured, it is complete
The estimation of paired harmonic wave interference, unknown nonlinear function and system mode；
It is for harmonic wave d design interference observers：
Wherein,W estimate is represented,RepresentFirst derivative,For y estimate,For estimating for harmonic wave interference d
Evaluation, L are the gain matrix of interference observer；
In order to eliminate the influence of unknown nonlinear function, it is defined as follows：
Then nonlinear system can be expressed as the form of augmented system：
Wherein, x1,x2,...,xn+1For the state of augmented system；
With reference to estimation of the interference observer to harmonic wave interferenceExtended state observer is designed to estimate unknown nonlinear letter
Number and system mode：
Wherein,State x is represented respectively1,x2,...,xn+1Estimate；For extended state observer
Output, represent y estimate；k1,k2,...,kn+1For observer gain；
The evaluated error for defining interference observer and extended state observer is Error dynamics, which can then be obtained, is：
Wherein, the rank matrix of (n+1) × 1N+1 rank square formations(n+1) × 2 rank square
Battle array(n+1) × 1 rank matrix1 × (n+1) rank Matrix C=[1 0 ... 0], 0 is the zero moment of appropriate dimension
Battle array.
3rd step, according to the output of interference observer and extended state observer design automatic disturbance rejection controller, based on point
Theoretical from theorem and POLE PLACEMENT USING, the gain for completing observer and controller solves；
Output based on interference observer and extended state observer designs the automatic disturbance rejection controller of following form：
Wherein, symbol ∑ represents summation operation, p1,p2,...,pnRepresent controller gain；
System dynamics equation is carried it into, can be obtained：
Based on linear system separation theorem, interference observer gain matrix L and extended state observer gain matrix K=
[k1 k2 … kn+1]TIt can be solved respectively by POLE PLACEMENT USING：
Wherein, s represents complex variable, and I represents the unit matrix of appropriate dimension, symbol | | represent to solve the ranks of square formation
Formula, ω0>0、ω1>0 is given constant, represents the bandwidth of system.In the implementation case, the value of each element in L is tried to achieve
Between -5 to 5, each element in K is between -100 to 100, parameter pi(i=1,2 ... value n) -10 to 10 it
Between.
The content not being described in detail in description of the invention belongs to prior art known to professional and technical personnel in the field.
Claims (4)
1. a kind of nonlinear system Auto-disturbance-rejection Control based on interference observer, it is characterised in that comprise the following steps：
The first step, harmonic wave interference and unknown nonlinear function are subjected to mathematical character, establish the external model of harmonic wave interference；
Second step, on the basis of mathematical character with modeling, based on control input and output information is measured, designs interference observer
And extended state observer, complete the estimation to harmonic wave interference, unknown nonlinear function and system mode；
3rd step, automatic disturbance rejection controller is designed according to the estimate of interference observer and extended state observer, based on separation
Theorem is theoretical with POLE PLACEMENT USING, and the gain for completing observer and controller solves.
2. the nonlinear system Auto-disturbance-rejection Control according to claim 1 based on interference observer, it is characterised in that：
The first step is implemented as follows：
Consider the single-input single-output nonlinear system containing harmonic wave interference and unknown nonlinear function as follows：
Wherein, n>0 is the order of system, and y represents model output, y(n)Y n order derivatives are represented,Represent that y single order is led
Number；f(y,y(1),...,y(n-1)) the unknown nonlinear function that single order can be led is represented, represent unknown nonlinear dynamic；B is known
Constant, u are control input, ymRepresent to measure output；Harmonic wave interference known to disturbing d expression frequency informations, is characterized asD withUnknown amplitude and phase are represented respectively, and ω is given frequency, and t represents the moment；
Harmonic wave interference d is described by following external model：
Wherein, w be external model state, coefficient matrixV=[1 0]；
Unknown nonlinear functionMeet single order can sliver part, i.e.,Wherein, h is
Unknown bounded function.
3. the nonlinear system Auto-disturbance-rejection Control according to claim 1 based on interference observer, it is characterised in that：
The second step is implemented as follows：
It is for harmonic wave interference d design interference observers
Wherein, w is the state of harmonic wave interference d external model,W estimate is represented,RepresentFirst derivative；Y is represented
The model output of nonlinear system, ymRepresent to measure output,For y estimate,For harmonic wave interference d estimate, L is interference
The gain matrix of observer；Coefficient matrixV=[1 0], ω are harmonic wave interference d given frequency；
In order to eliminate the influence of unknown nonlinear function, it is defined as follows：
Wherein, n>0 be system order, y(n)Y n order derivatives are represented,Represent y first derivative, f (y, y(1),...,y(n-1))
Represent the unknown nonlinear function that single order can be led；
Then nonlinear system is expressed as the form of augmented system：
Wherein, xiFor the state of augmented system, i=1,2 ..., n+1；B is the known constant in nonlinear system, and u is control
System input,For unknown bounded function；
With reference to estimation of the interference observer to harmonic wave interferenceIt is as follows to design extended state observer：
Wherein,Expression state xiEstimate,For the output of extended state observer, y estimate is represented；kiTo expand shape
State observer gain, i=1,2 ..., n+1.
4. the nonlinear system Auto-disturbance-rejection Control according to claim 1 based on interference observer, it is characterised in that：
3rd step is implemented as follows：
Output based on interference observer and extended state observer designs the automatic disturbance rejection controller of following form：
Wherein, symbol ∑ represents summation operation, p1,p2,...,pnRepresent controller gain；B is known in nonlinear system
Constant, n>0 is the order of system；xiWith xn+1For the state of augmented system, i represents the integer sequence from 1 to n,WithRespectively
Expression state xiWith xn+1Estimate,Represent harmonic wave interference d estimate；
Based on linear system separation theorem, interference observer gain matrix L and extended state observer gain matrix K=[k1 k2
… kn+1]TAnd controller gain p1,p2,...,pnSolved respectively by POLE PLACEMENT USING：
Wherein, n+1 ranks square formation(n+1) × 2 rank matrix1 × (n+1) rank Matrix C
=[1 0 ... 0], matrixV=[1 0], ω are harmonic wave interference d given frequency；S represents complex variable, I
The unit matrix of appropriate dimension is represented, symbol | | represent to solve the determinant of square formation, ω0>0、ω1>0 constant to give, table
Show the bandwidth of system.
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| CN112803461B (en) * | 2021-03-04 | 2023-04-28 | 上海海事大学 | Active disturbance rejection control method of battery energy storage converter based on extended state observer |
| CN113050493B (en) * | 2021-03-19 | 2022-03-04 | 大连理工大学 | Output feedback control method for inverted pendulum system of trolley in networked environment |
| CN113419462B (en) * | 2021-07-08 | 2022-06-10 | 北京航空航天大学杭州创新研究院 | Power grid current composite prediction control method based on harmonic interference observer |
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