CN107168072B - A kind of non-matching interference system Auto-disturbance-rejection Control based on interference observer - Google Patents
A kind of non-matching interference system Auto-disturbance-rejection Control based on interference observer Download PDFInfo
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Abstract
The present invention relates to a kind of non-matching interference system Auto-disturbance-rejection Control based on interference observer, and matching and the interference of non-matching two rahmonic and unknown nonlinear function are carried out into mathematical character；Interference observer is separately designed to the interference of two rahmonics, completes the real-time estimation to harmonic wave interference；Output design extended state observer based on interference observer, completes the estimation to unknown nonlinear function and system mode；Next, with reference to the estimate of non-matching interference, coordinate transform is completed by introducing new state variable；Automatic disturbance rejection controller is designed according to the output of interference observer and extended state observer on the basis of above-mentioned Coordinate Conversion；Theoretical based on separation theorem and POLE PLACEMENT USING, the gain for completing observer and controller solves, so as to complete the design of controller；The present invention has the advantages that strong antijamming capability, control accuracy are high, available for the high-precision control containing matching with non-matching harmonic wave interference and Unknown Nonlinear Systems.
Description
Technical Field
The invention relates to an active disturbance rejection control method of a non-matching disturbance system based on a disturbance observer, which can realize simultaneous estimation and cancellation of matching and non-matching harmonic disturbance and an unknown nonlinear function and can be used for system control containing harmonic disturbance and the unknown nonlinear function.
Background
Due to the complexity of the controlled object and the task, unknown nonlinear factors of various sources such as modeling errors, parameter changes, input nonlinearity and the like have serious influence on the performance of the control system, and even the system is diverged. In addition, the control performance is further deteriorated by interference factors from the external environment, internal sensors, actuators, and the like. In response to the unknown non-linear factors and interference, the scholars propose various advanced control methods, such as LQG control, PID control and H ∞ Control, and the like. However, the LQG optimal control theory is designed based on a model of a system, has a high degree of dependence on the model, is limited to a system affected by white gaussian noise, and cannot guarantee performance when unknown nonlinearity or other types of interference exist in the system. The PID control generated in the twenties of the last century has been dominant in industrial control so far due to its advantages of simple structure, independence from system models, and the like. However,PID control also has its limitations: firstly, PID control completely ignores the information of the system model; secondly, differential signals in PID control are often difficult to obtain well, and high-frequency noise is easy to generate; thirdly, the integration link brings about phase lag, oscillation and other consequences; finally, parameter adjustment of PID control is complicated; in addition, the PID control can only compensate for constant interference, and has poor inhibition capability on harmonic waves and unknown nonlinear factors. H ∞ And the robust control mode can only carry out interference suppression on harmonics and unknown nonlinear functions, and cannot compensate, so that the control precision is limited and the conservation is high.
In order to improve the control performance and compensate various disturbance factors suffered by a system, korean Jingqing teaching provides an Active Disturbance Rejection Control (ADRC) method with disturbance compensation capability from PID control, which comprises a tracking differentiator, an extended state observer and a nonlinear feedback controller, and can convert a complex nonlinear system into a serial integral type form, realize the real-time estimation and compensation of unknown nonlinear functions and various disturbance factors, and overcome the limitation that the modern control theory excessively depends on a system model. However, since the conventional ADRC ignores the model of the interference and estimates and compensates all the disturbances and unknown non-linearities as the total disturbance with bounded derivatives, the estimation effect of the conventional ADRC on the harmonic interference is not ideal, for example, patents with patent publication nos. ZL200410070983.2 and 201510359468.4 adopt the active disturbance rejection control method to treat all the disturbances and non-linearities as the total disturbance, but the model and accurate estimation research on the harmonic interference is lacked.
The control (DOBC) based on the interference observer fully utilizes model information of the interference, can realize accurate estimation and compensation of the interference such as harmonic waves, constant values and the like, can be conveniently combined with other controls, and realizes simultaneous suppression and compensation of a plurality of interferences through composite control, for example, patents with patent authorizations of ZL200910086897.3 and ZL 201081167.0 adopt a composite control mode to realize simultaneous compensation and suppression of various disturbances such as harmonic waves and the like. However, the current composite control has two limitations: firstly, under consideration of harmonic interference, the unknown nonlinear dynamics is not considered sufficiently; secondly, most of the considered interference is matched interference, and simple and effective compensation modes such as state feedback lack for the situation that non-matched interference and matched interference exist simultaneously are researched. Many practical systems often contain non-matching harmonic interference, such as aircraft, permanent magnet synchronous motors, magnetic levitation control systems, and the like. The problem of cancellation of non-matching harmonic interference has been one of the difficulties of research, since it is not in the control channel.
In summary, the interference compensation research on the multi-source interference system including both matched and unmatched harmonic interference and unknown nonlinear dynamics is still seriously insufficient. Because the interference and the unknown nonlinear dynamics are mixed and coupled with each other, the interference estimation error and the nonlinear dynamics estimation error are influenced with each other, the interference is from different control channels, no research about effective combination of DOBC and ADRC is found at present, respective advantages of DOBC and ADRC need to be fully combined, simultaneous cancellation of various interferences and nonlinear dynamics is realized, and therefore system accuracy and robustness are enhanced.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: aiming at the problem that the existing control method is difficult to compensate interference, particularly to compensate non-matching harmonic interference and unknown nonlinear functions at the same time, the interference observer-based active disturbance rejection control method with non-matching and matching harmonic interference and unknown nonlinear function real-time estimation and cancellation capabilities is provided, has the advantages of strong anti-interference capability, high control precision and the like, and can be used for high-precision control of a system containing matching and non-matching harmonic interference and unknown nonlinear functions.
The technical solution of the invention is as follows: a non-matching interference system active disturbance rejection control method based on a disturbance observer is provided, and aiming at a non-linear system containing non-matching and matching harmonic interference and an unknown non-linear function, firstly, mathematical characterization is carried out on two types of matching and non-matching harmonic interference and the unknown non-linear function; secondly, designing interference observers for the two types of harmonic interference respectively to complete real-time estimation of the harmonic interference; thirdly, designing an extended state observer based on the output of the disturbance observer, and finishing the estimation of the unknown nonlinear function and the system state; then, combining the estimated value of the non-matching interference, and finishing coordinate transformation by introducing a new state variable; further, an active disturbance rejection controller is designed according to the output of the disturbance observer and the extended state observer on the basis of the coordinate conversion; finally, based on the separation theorem and the pole allocation theory, gain solution of the observer and the controller is completed, and therefore design of the controller is completed; the specific implementation steps are as follows:
(1) Performing mathematical characterization on the two types of harmonic interference of matching and non-matching and an unknown nonlinear function:
consider a second order system with matched and unmatched harmonic interference and an unknown nonlinear function as follows:
wherein x is 1 And x 2 In order to be in the state of the system,and withIs the time derivative of the system state, y is the measurement output, x = [ x ] 1 x 2 ] T U is the control input, b is a constant greater than zero, f (x) 1 ,x 2 ) Is a first order derivable unknown non-linear function; d 0 And d 1 Non-matching harmonic interference and matching harmonic interference, respectively, that indicate that frequency information is known, can be characterized asWherein D is 0 And D 1 Which represents the magnitude of the unknown signal,and withRepresenting unknown phase, ω 0 And omega 1 Representing a known frequency, t represents a time instant;
non-matching harmonic interference d 0 And matched harmonic interference d 1 Can be described by the following external models, respectively:
w and xi are states of external model, coefficient matrixV 0 ＝V 1 ＝[1 0]；
Unknown non-linear function f (x) 1 ,x 2 ) Satisfying a first order conductibility condition, i.e.Wherein h is an unknown bounded function;
(2) Designing interference observers for the two types of harmonic interference respectively to complete real-time estimation of the harmonic interference:
for non-matching harmonic interference d 0 Designing a disturbance observer as follows:
wherein, the first and the second end of the pipe are connected with each other,denotes d 0 Is determined by the estimated value of (c),an estimate of the state w is represented,v 0 for auxiliary state variables, L 1 Is an observer gain matrix;
for matching harmonic interference d 1 Designing a disturbance observer as follows:
wherein the content of the first and second substances,denotes d 1 Is determined by the estimated value of (c),estimate value representing ξ, order state x 3 ＝f(x 1 ,x 2 ) To do soIs in a state x 3 Estimated value of v 1 For auxiliary state variables, L 2 Is an observer gain matrix;
(3) Designing an extended state observer based on the output of the disturbance observer, and finishing the estimation of an unknown nonlinear function and a system state:
x is to be 3 As an augmented state, a second order system ∑ 0 Can be written as a form of augmented system:
based on interference observer sigma 3 And sigma 4 Output of (2) to the augmentation System ∑ 5 The extended state observer is designed as follows:
wherein, the first and the second end of the pipe are connected with each other,respectively represent the state x 1 ,x 2 ,x 3 Is determined by the estimated value of (c),an estimate of the value of y is represented,denotes the estimated value of x,/ 1 ,l 2 ,l 3 Represents the gain of the extended state observer;
incorporating an external model ∑ 1 And disturbance observer sigma 3 Non-matching harmonic interference d can be obtained 0 Estimation errorDynamic equation of (c):
incorporating an external model ∑ 2 And disturbance observer sigma 4 Can obtain the matched harmonic interference d 1 Estimation errorDynamic equation of (c):
wherein the content of the first and second substances,represents a state x 3 The estimation error of (2);
similarly, combine the augmentation system ∑ 5 And extended state observer Σ 6 Obtaining a state estimation errorDynamic equation of i =1,2,3:
the three types of dynamic equations are combined and correspondingly transformed, so that the following can be obtained:
wherein the content of the first and second substances,the specific expression of the coefficient matrix is as follows:
C 1 ＝[0 0 1]，
(4) And combining the estimated value of the non-matching interference, and finishing coordinate transformation by introducing a new state variable:
based on non-matching harmonic interference d 0 And matched harmonic interference d 1 Estimated value of (1), second order system ∑ 0 Can be converted into:
wherein the content of the first and second substances,
ignoring interference and state estimation errors and introducing a new state variable z 1 ＝x 1 ，z 3 ＝x 3 Can be obtained as followsThe transformed control system:
(5) Designing an active disturbance rejection controller according to the output of the disturbance observer and the extended state observer on the basis of the coordinate conversion:
for system ∑ 9 Designing an active disturbance rejection controller based on a disturbance observer as follows:
wherein p is 1 ,p 2 In order to control the gain of the controller,
(6) Based on the separation theorem and the pole allocation theory, the gain solution of the observer and the controller is completed, so that the design of the controller is completed:
based on the linear system separation theorem, the gain matrix L of the disturbance observer 1 And L 2 Extended state observer gain matrix L and controller gain p 1 ,p 2 The solution can be solved by the pole configuration:
where s represents a complex variable, I represents an identity matrix of appropriate dimensions, the symbol | · | represents the determinant for solving the square matrix, ω 0 >0、ω 1 &And gt, 0 is a given constant and represents the bandwidth of the system.
Compared with the prior art, the invention has the advantages that: the invention completes the estimation and compensation of two types of harmonic interference by means of the interference observer, particularly solves the estimation and compensation problem of non-matching harmonic interference by coordinate transformation, is combined with the extended state observer, completes the simultaneous estimation and compensation problem of matching and non-matching harmonic interference and unknown nonlinear dynamics, overcomes the limitation that a single active disturbance rejection controller is difficult to compensate harmonic interference, and can be used for high-precision control of a system containing matching and non-matching harmonic interference and unknown nonlinear system.
Drawings
Fig. 1 is a flow chart of an active disturbance rejection control method of a non-matching disturbance system based on a disturbance observer.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples.
As shown in fig. 1, the method of the invention comprises the following steps:
firstly, performing mathematical characterization on two types of harmonic interference of matching and non-matching and an unknown nonlinear function:
consider a second order system with matched and unmatched harmonic interference and an unknown nonlinear function as follows:
wherein x is 1 And x 2 In order to be in the state of the system,andis the time derivative of the system state, y is the measurement output, x = [ x ] 1 x 2 ] T U is the control input, b is a constant greater than zero, f (x) 1 ,x 2 ) Is a first order derivable unknown non-linear function; d is a radical of 0 And d 1 Non-matching harmonic interference and matching harmonic interference, respectively, that indicate that frequency information is known, can be characterized asWherein D is 0 And D 1 Which represents the magnitude of the unknown signal,andrepresenting unknown phase, ω 0 And omega 1 Representing a known frequency, t represents a time instant; in the embodiment of the invention, an unknown nonlinear function is taken asTake b =1,D 0 ＝D 1 ＝0.05，ω 0 ＝10，ω 1 ＝15；
Non-matching harmonic interference d 0 And matching harmonic interference d 1 Can be described by the following external models, respectively:
w and xi are states of external model, coefficient matrixV 0 ＝V 1 ＝[1 0]；
Unknown non-linear function f (x) 1 ,x 2 ) Is full ofSufficient for a first order conduction condition, i.e.Wherein h is an unknown bounded function;
secondly, designing interference observers for the two types of harmonic interference respectively to complete real-time estimation of the harmonic interference:
for non-matched harmonic interference d 0 Designing a disturbance observer as follows:
wherein, the first and the second end of the pipe are connected with each other,denotes d 0 Is determined by the estimated value of (c),an estimate value, v, representing the state w 0 For auxiliary state variables, L 1 An observer gain matrix;
for matching harmonic interference d 1 Designing a disturbance observer as follows:
wherein the content of the first and second substances,denotes d 1 Is determined by the estimated value of (c),estimate value representing ξ, order state x 3 ＝f(x 1 ,x 2 ) To is thatIs a state x 3 Estimated value of v 1 For auxiliary state variables, L 2 Is the gain moment of the observerArraying;
thirdly, designing an extended state observer based on the output of the disturbance observer, and finishing the estimation of the unknown nonlinear function and the system state:
x is to be 3 As an augmented state, a second order system ∑ 0 Can be written as a form of augmented system:
based on interference observer sigma 3 And sigma 4 Output of (2) to the augmentation System ∑ 5 The extended state observer is designed as follows:
wherein the content of the first and second substances,respectively represent the state x 1 ,x 2 ,x 3 Is determined by the estimated value of (c),an estimate of the value of y is represented,denotes the estimated value of x,/ 1 ,l 2 ,l 3 Represents the gain of the extended state observer;
incorporating an external model ∑ 1 And interference observer sigma 3 Non-matching harmonic interference d can be obtained 0 Estimation errorThe dynamic equation of (c):
incorporating an external model ∑ 2 And interference observer sigma 4 Can obtain the matched harmonic interference d 1 Estimation errorDynamic equation of (c):
wherein the content of the first and second substances,represents a state x 3 The estimated error of (2);
similarly, combine the augmentation system ∑ 5 And extended state observer Σ 6 Obtaining a state estimation errorDynamic equation of i =1,2,3:
the three dynamic equations are combined and correspondingly transformed, so that the following can be obtained:
wherein the content of the first and second substances,the specific expression of the coefficient matrix is as follows:
C 1 ＝[0 0 1]，
and fourthly, combining the estimated value of the non-matching interference, and finishing coordinate transformation by introducing a new state variable:
based on non-matching harmonic interference d 0 And matched harmonic interference d 1 Estimated value of (1), second order system ∑ 0 Can be converted into:
wherein the content of the first and second substances,
ignoring interference and state estimation errors and introducing a new state variable z 1 ＝x 1 ，z 3 ＝x 3 A control system can be obtained after transformation as follows:
fifthly, designing an active disturbance rejection controller according to the output of the disturbance observer and the extended state observer on the basis of the coordinate conversion:
for system ∑ 9 Designing an active disturbance rejection controller based on a disturbance observer as follows:
wherein p is 1 ,p 2 In order to control the gain of the controller,
and sixthly, completing gain solution of the observer and the controller based on a separation theorem and a pole allocation theory, thereby completing the design of the controller:
based on the separation theorem of linear system, the gain matrix L of the disturbance observer 1 And L 2 Extended state observer gain matrix L and controller gain p 1 ,p 2 The solution can be solved by the pole configuration:
where s represents a complex variable, I represents a unit matrix of appropriate dimensions, the symbol | · | represents a determinant for solving a square matrix, and ω represents a complex variable 0 >0、ω 1 &And gt, 0 is a given constant and represents the bandwidth of the system. In the present embodiment, the value of each element in L, K is found to be between-5 and 5, and the parameter p 1 ,p 2 Is between-20 and 20.
Those matters not described in detail in the present specification are well known in the art to which the skilled person pertains.
Claims (7)
1. A non-matching interference system active disturbance rejection control method based on a disturbance observer is characterized by comprising the following steps:
firstly, performing mathematical representation on two types of harmonic interference of matching and non-matching and an unknown nonlinear function;
secondly, designing interference observers for the two types of harmonic interference of matching and non-matching respectively to complete real-time estimation of the harmonic interference;
thirdly, designing an extended state observer based on the output of the disturbance observer, and finishing the estimation of an unknown nonlinear function and a system state;
fourthly, combining the estimated value of the non-matching interference, and finishing coordinate transformation by introducing a new state variable;
fifthly, designing an active disturbance rejection controller according to the output of the disturbance observer and the extended state observer on the basis of the coordinate conversion of the fourth step;
and sixthly, finishing the gain solution of the observer and the controller based on the separation theorem and the pole allocation theory, thereby finishing the design of the controller.
2. The disturbance observer-based active disturbance rejection control method of the non-matching disturbance system according to claim 1, wherein: the first step is specifically realized as follows:
consider a second order system with matched and unmatched harmonic interference and an unknown nonlinear function as follows:
wherein x is 1 And x 2 In order to be in the state of the system,andis the time derivative of the system state, y is the measurement output, x = [ x ] 1 x 2 ] T U is the control input, b is a constant greater than zero, f (x) 1 ,x 2 ) Is a first order derivable unknown non-linear function; d is a radical of 0 And d 1 Respectively represent the non-matching harmonic interference and the matching harmonic interference of which the frequency information is known and are characterized byD 0 And D 1 Which represents the magnitude of the unknown signal,and withRepresenting unknown phase, ω 0 And omega 1 Representing a known frequency, t representing a time instant;
non-matching harmonic interference d 0 And matched harmonic interference d 1 Described by the following external models, respectively:
w and xi are states of external model, coefficient matrix V 0 ＝V 1 ＝[1 0]；
Unknown non-linear function f (x) 1 ,x 2 ) Satisfying a first order conductibility condition, i.e.Where h is an unknown bounded function.
3. The disturbance observer-based non-matching disturbance system active disturbance rejection control method according to claim 1, wherein: the second step is specifically realized as follows:
for non-matching harmonic interference d 0 Design the disturbance observer as
Wherein x is 1 And x 2 For a second order system ∑ 0 W is the external model Σ 1 In the state of (a) to (b),denotes d 0 Is determined by the estimated value of (c),an estimate value, v, representing the state w 0 For auxiliary state variables, L 1 In order to be an observer gain matrix,and V 0 ＝[1 0]All coefficient matrices, ω, being known 0 For non-matching harmonic interference d 0 The known frequency of (d);
for matching harmonic interference d 1 Design the disturbance observer as
Wherein u is a second order system ∑ 0 B is a second order system sigma 0 Medium constant greater than zero, ξ is the external model ∑ 2 In the state of (a) to (b),denotes d 1 Is determined by the estimated value of (c),an estimate representing ξ; let state x 3 ＝f(x 1 ,x 2 ) To do soIs a state x 3 Estimated value of v 1 For auxiliary state variables, L 2 In order to be an observer gain matrix,and V 1 ＝[1 0]All coefficient matrices, ω, being known 1 For matching harmonic interference d 1 Is known.
4. The disturbance observer-based non-matching disturbance system active disturbance rejection control method according to claim 1, wherein: the third step is specifically realized as follows:
x is to be 3 As an augmented state, a second order system ∑ 0 Writing is in the form of an augmentation system:
wherein x is 1 And x 2 For a second order system ∑ 0 System state of (1), x 3 ＝f(x 1 ,x 2 ) Is an unknown non-linear function, u is a control input, b is a constant greater than zero, d 0 And d 1 Respectively representing the known non-matching harmonic interference and the matching harmonic interference of frequency information, h is an unknown bounded function, and x = [ x ] 1 x 2 ] T Y is the measurement output;
based on interference observer sigma 3 And sigma 4 Output of (2) to the augmentation System ∑ 5 The extended state observer is designed as follows:
wherein the content of the first and second substances,respectively represent the state x 1 ,x 2 ,x 3 Is determined by the estimated value of (c),an estimate of the value of y is represented,an estimate of the value of x is represented,denotes d 0 Is determined by the estimated value of (c),denotes d 1 Estimate of (a), l 1 ,l 2 ,l 3 Representing the gain of the extended state observer;
incorporating an external model ∑ 1 And disturbance observer sigma 3 Obtaining an unmatched harmonic interference d 0 Estimation errorWherein w is the external model ∑ 1 In the state of (a) or (b),an estimate representing state w; incorporating an external model ∑ 2 And disturbance observer sigma 4 Obtaining the matching harmonic interference d 1 Estimation errorWhere ξ is the external model Σ 2 In the state of (a) or (b),an estimate representing ξ; combined augmentation System ∑ 5 And extended state observer Σ 6 Obtaining a state estimation errorI =1,2,3; combining the dynamic equations to obtain:
wherein, the first and the second end of the pipe are connected with each other,the specific expression of the coefficient matrix is as follows:
C 1 ＝[0 0 1]， V 0 ＝[1 0]，V 1 ＝[1 0]；L 1 for observer ∑ 3 Of the gain matrix, L 2 For observer ∑ 4 Of the gain matrix omega 0 For non-matching harmonic interference d 0 Of a known frequency, ω 1 For matching harmonic interference d 1 Is known.
5. The disturbance observer-based non-matching disturbance system active disturbance rejection control method according to claim 1, wherein: the fourth step is specifically realized as follows:
based on non-matching harmonic interference d 0 And matched harmonic interference d 1 Estimated value of (1), second order system ∑ 0 Is converted into
Wherein x is 1 And x 2 For a second order system ∑ 0 Y is a measurement output, x = [ x ] 1 x 2 ] T U is the control input, b is a constant greater than zero, f (x) 1 ,x 2 ) Is a first order derivable unknown non-linear function;denotes d 0 Is determined by the estimated value of (c),denotes d 1 Is determined by the estimated value of (c),andrespectively represents d 0 And d 1 The estimated error of (2);
neglecting the estimation error of interference and state, combining the estimated value of non-matching interferenceAnd introduces a new state variable z 1 ＝x 1 ，z 3 ＝x 3 Obtaining a control system after transformation as follows:
wherein x is 3 ＝f(x 1 ,x 2 ) For unknown non-linear functions, w is the external model ∑ 1 In the state of (a) to (b),an estimate of the state w is represented,and V 0 ＝[10]All coefficient matrices, ω, being known 0 For non-matching harmonic interference d 0 H is an unknown bounded function.
6. The disturbance observer-based active disturbance rejection control method of the non-matching disturbance system according to claim 1, wherein: the fifth step is specifically realized as follows:
according to the control system ∑ 9 Designing an active disturbance rejection controller based on a disturbance observer as follows:
wherein p is 1 ,p 2 B is a constant greater than zero for the controller gain,representation extended state observer Σ 6 An estimate of the state variable of (d) c, an estimate of the state w is represented by,denotes d 0 Is determined by the estimated value of (c),denotes d 1 Is determined by the estimated value of (c),and V 0 ＝[1 0]All coefficient matrices, ω, being known 0 For non-matching harmonic interference d 0 Is known.
7. The disturbance observer-based non-matching disturbance system active disturbance rejection control method according to claim 1, wherein: the sixth step is specifically realized as follows:
gain matrix L of disturbance observer based on linear system separation theorem 1 And L 2 Gain matrix L of extended state observer and controller gain p 1 ,p 2 Solving by pole allocation respectively:
where s denotes a complex variable, I denotes an identity matrix of appropriate dimensions, the symbol | · | denotes the determinant for solving the square matrix,a given constant, representing the bandwidth of the system; p is a radical of 1 ,p 2 In order to control the gain of the controller,C 1 ＝[001]， V 0 ＝[1 0]，V 1 ＝[1 0]；L 1 for observer ∑ 3 Of the gain matrix, L 2 For observer ∑ 4 B is a constant greater than zero, l 1 ,l 2 ,l 3 Representation extended state observer Σ 6 Gain of (a), omega 0 For non-matching harmonic interference d 0 Of a known frequency, ω 1 For matching harmonic interference d 1 Is known.
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