CN107168072A - A kind of non-matching interference system Auto-disturbance-rejection Control based on interference observer - Google Patents

Abstract

The invention relates to a non-matching disturbance system ADRC control method based on a disturbance observer, which mathematically characterizes two types of harmonic disturbances of matching and non-matching and unknown nonlinear functions; and designs disturbance observers for the two types of harmonic disturbances respectively , to complete the real-time estimation of harmonic interference; design an extended state observer based on the output of the disturbance observer, and complete the estimation of the unknown nonlinear function and system state; The coordinate transformation of the variable is completed; on the basis of the above coordinate transformation, the ADRC controller is designed according to the output of the disturbance observer and the extended state observer; based on the separation theorem and the pole configuration theory, the gain solution of the observer and the controller is completed, thereby completing The design of the controller; the invention has the advantages of strong anti-interference ability and high control precision, and can be used for high-precision control of matching and non-matching harmonic interference and unknown nonlinear systems.

Description

An Active Disturbance Rejection Control Method for Unmatched Disturbance Systems Based on Disturbance Observer

technical field

The invention relates to a disturbance observer-based active disturbance rejection control method for non-matching disturbance systems, which can realize simultaneous estimation and cancellation of matching and non-matching harmonic disturbances and unknown nonlinear functions, and can be used for harmonic disturbances and unknown nonlinear functions function in system control.

Background technique

Due to the complexity of the controlled object and task, unknown nonlinear factors from various sources such as modeling error, parameter change, and input nonlinearity have a serious impact on the performance of the control system, and even cause the system to diverge. In addition, interference factors from the external environment, internal sensors and actuators further aggravate the deterioration of control performance. In view of the simultaneous existence of unknown nonlinear factors and disturbances, scholars have proposed many advanced control methods, such as LQG control, PID control, and H _∞ control, etc. However, the LQG optimal control theory is designed based on the model of the system, which is highly dependent on the model, and is limited to the system affected by Gaussian white noise, and the performance cannot be guaranteed when the system has unknown nonlinearity or other types of interference. The PID control, which was born in the 1920s, has been in a dominant position in industrial control so far because of its simple structure and no dependence on system models. However, PID control also has its limitations: first, PID control completely ignores the information of the system model; second, the differential signal in PID control is often difficult to obtain better, and it is easy to generate high-frequency noise; third, the integration link brings phase lag And oscillation and other consequences; finally, the parameter adjustment of PID control is more cumbersome; in addition, PID control can only compensate for constant value interference, and its ability to suppress harmonics and unknown nonlinear factors is poor. Robust control methods such as H _∞ can only suppress the interference of harmonics and unknown nonlinear functions, but cannot compensate, resulting in limited control accuracy and high conservatism.

In order to improve the control performance and compensate the various disturbance factors to the system, Professor Han Jingqing proposed the Active Disturbance Rejection Control (ADRC) method with disturbance compensation capability starting from PID control, including tracking differentiator, extended state observer and nonlinear feedback control Composed of three parts, the complex nonlinear system can be converted into a series-integral form, realizing real-time estimation and compensation of unknown nonlinear functions and various disturbance factors, and overcoming the limitation of modern control theory that relies too much on system models sex. However, because the traditional ADRC ignores the interference model, it estimates and compensates all disturbances and unknown nonlinearities as total disturbances with bounded derivatives, resulting in unsatisfactory estimation of harmonic interference. For example, patent authorization The patents No. ZL200410070983.2 and Application No. 201510359468.4 both adopt the active disturbance rejection control method, which treats all disturbances and nonlinearities as total disturbances, but lacks research on modeling and accurate estimation of harmonic disturbances.

Disturbance observer-based control (DOBC) makes full use of the disturbance model information, can realize accurate estimation and compensation of disturbances such as harmonics and constant values, and can be easily combined with other controls to realize multiple disturbances through compound control Simultaneous suppression and compensation. For example, the patents with patent authorization numbers ZL200910086897.3 and ZL201310081167.0 both use a composite control method to realize simultaneous compensation and suppression of multiple disturbances such as harmonics. However, the current composite control also has two limitations: First, while considering the harmonic interference, the unknown nonlinear dynamics are not considered enough; There is a lack of research on simple and effective compensation methods such as state feedback. However, many practical systems often contain non-matching harmonic interference, such as aircraft, permanent magnet synchronous motors, magnetic levitation control systems, and so on. Since it is not in the control channel, the cancellation of non-matching harmonic interference has always been one of the difficulties in research.

In summary, the research on interference compensation for multi-source interference systems with both matched and unmatched harmonic interference and unknown nonlinear dynamics is still seriously insufficient. Due to the mutual mixing and coupling of interference and unknown nonlinear dynamics, the interference estimation error and nonlinear dynamic estimation error affect each other, and the interference comes from different control channels. At present, no research on the effective combination of DOBC and ADRC has been found, and it is necessary to fully combine DOBC and ADRC With their respective advantages, the simultaneous cancellation of multiple disturbances and nonlinear dynamics can be achieved, thereby enhancing the accuracy and robustness of the system.

Contents of the invention

The technical solution of the present invention is to solve the problem that it is difficult to compensate the interference in the existing control method, especially the problem that it is difficult to compensate the non-matching harmonic interference and the unknown nonlinear function at the same time, and to provide a non-matching and matching harmonic The ADRC method based on the disturbance observer based on the real-time estimation and cancellation ability of disturbance and unknown nonlinear function has the advantages of strong anti-disturbance ability and high control precision, and can be used for systems with matched and unmatched harmonic disturbances and unknown nonlinear systems. High precision control.

The technical solution of the present invention is: an ADRC control method for a non-matching disturbance system based on a disturbance observer. For a nonlinear system containing non-matching and matching harmonic disturbances and unknown nonlinear functions, firstly, the matching and non-matching Match two types of harmonic interference and unknown nonlinear function for mathematical characterization; secondly, design interference observers for the two types of harmonic interference to complete real-time estimation of harmonic interference; thirdly, design extended state observation based on the output of the interference observer , to complete the estimation of the unknown nonlinear function and the system state; next, combine the estimated value of the non-matching disturbance, and complete the coordinate transformation by introducing a new state variable; further, on the basis of the above coordinate transformation, according to the disturbance observer and The ADRC controller is designed based on the output of the extended state observer; finally, based on the separation theorem and the pole configuration theory, the gain solution of the observer and the controller is completed, thereby completing the design of the controller; the specific implementation steps are as follows:

(1) Mathematically characterize two types of harmonic interference, matching and non-matching, and unknown nonlinear functions:

Consider the following second-order system with matched and unmatched harmonic disturbances and an unknown nonlinear function:

Among them, x ₁ and x ₂ are the system state, and is 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 the first-order derivable unknown Non-linear function; d ₀ and d ₁ represent the non-matching harmonic interference and matching harmonic interference with known frequency information respectively, which can be characterized as Among them, D ₀ and D ₁ represent unknown amplitudes, and Represents the unknown phase, ω ₀ and ω ₁ represent the known frequency, t represents the time;

The unmatched harmonic interference d ₀ and the matched harmonic interference d ₁ can be described by the following external models respectively:

Among them, w and ξ are the state of the external model, and the coefficient matrix V ₀ =V ₁ =[1 0];

The unknown nonlinear function f(x ₁ ,x ₂ ) satisfies the first-order derivability condition, namely Among them, h is an unknown bounded function;

(2) Design disturbance observers for the two types of harmonic disturbances to complete the real-time estimation of harmonic disturbances:

The disturbance observer designed for the unmatched harmonic disturbance d ₀ is:

in, represents the estimated value of _d0 , Represents the estimated value of the state w, v ₀ is the auxiliary state variable, L ₁ is the observer gain matrix;

The disturbance observer designed for the matched harmonic disturbance d ₁ is:

in, represents the estimated value _of d1, represents the estimated value of ξ, let the state x ₃ =f(x ₁ ,x ₂ ), and is the estimated value of state x ₃ , v ₁ is the auxiliary state variable, L ₂ is the observer gain matrix;

(3) Based on the output of the disturbance observer, an extended state observer is designed to complete the estimation of the unknown nonlinear function and system state:

Taking x ₃ as the augmented state, the second-order system Σ ₀ can be written in the form of the augmented system:

Based on the output of disturbance observers Σ ₃ and Σ ₄ , the extended state observer for the augmented system Σ ₅ is designed as:

in, represent the estimated values of states x ₁ , x ₂ , and x ₃ respectively, represents the estimated value of y, Represents the estimated value of x, l ₁ , l ₂ , l ₃ represent the gain of the expanded state observer;

Combining the external model Σ ₁ and the disturbance observer Σ ₃ , the estimation error of the mismatched harmonic disturbance d ₀ can be obtained The dynamic equation of :

Combining the external model Σ ₂ with the disturbance observer Σ ₄ , the estimation error of the matched harmonic disturbance d ₁ can be obtained The dynamic equation of :

in, Indicates the estimation error of state x ₃ ;

Similarly, combining the augmented system Σ ₅ and the extended state observer Σ ₆ , the state estimation error is obtained Dynamic equations for i=1,2,3:

Combining the above three types of dynamic equations and performing corresponding transformations, we can get:

in, The specific expression of the coefficient matrix is as follows:

C ₁ =[0 0 1],

(4) Combined with the estimated value of non-matching interference, the coordinate transformation is completed by introducing a new state variable:

Based on the estimated values of unmatched harmonic disturbance d ₀ and matched harmonic disturbance d ₁ , the second-order system Σ ₀ can be transformed into:

in,

Neglect the estimation error of interference and state and introduce a new state variable z ₁ =x ₁ , z ₃ ＝x ₃ , the following transformed control system can be obtained:

(5) On the basis of the above-mentioned coordinate transformation, the ADRC controller is designed according to the output of the disturbance observer and the extended state observer:

For the system Σ ₉ , the ADRC based on the disturbance observer is designed as:

Among them, p ₁ and p ₂ are controller gains,

(6) Based on the separation theorem and pole allocation theory, the gain solution of the observer and the controller is completed, so as to complete the design of the controller:

Based on the linear system separation theorem, the disturbance observer gain matrices L ₁ and L ₂ , the extended state observer gain matrix L and the controller gains p ₁ and p ₂ can be solved by pole configuration respectively:

Among them, s represents a complex variable, I represents an identity matrix of appropriate dimension, the symbol |·| represents the determinant of solving a square matrix, ω ₀ >0, ω ₁ >0 are given constants, which represent the bandwidth of the system.

Compared with the prior art, the present invention has the advantages of: the present invention completes the estimation and compensation of two types of harmonic interference by means of the disturbance observer, especially solves the problem of estimation and compensation of non-matching harmonic interference through coordinate transformation, and Combined with the extended state observer, the problem of simultaneously estimating and compensating matching and non-matching harmonic disturbances and unknown nonlinear dynamics is accomplished, overcoming the limitation that a single active disturbance rejection controller is difficult to compensate harmonic disturbances, and can be used for High-precision control of systems with unmatched harmonic disturbances and unknown nonlinearities.

Description of drawings

Fig. 1 is a flow chart of an ADRC control method for a non-matching disturbance system based on a disturbance observer.

detailed description

The present invention will be described in further detail below in conjunction with the accompanying drawings and embodiments.

As shown in Figure 1, the specific implementation steps of the present invention are as follows:

The first step is to mathematically characterize the matching and non-matching two types of harmonic interference and the unknown nonlinear function:

Consider the following second-order system with matched and unmatched harmonic disturbances and an unknown nonlinear function:

Among them, x ₁ and x ₂ are the system state, and is 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 the first-order derivable unknown Non-linear function; d ₀ and d ₁ represent the non-matching harmonic interference and matching harmonic interference with known frequency information respectively, which can be characterized as Among them, D ₀ and D ₁ represent unknown amplitudes, and represents the unknown phase, ω ₀ and ω ₁ represent the known frequency, and t represents the moment; in the embodiment of the present invention, the unknown nonlinear function is Take b=1, D ₀ =D ₁ =0.05, ω ₀ =10, ω ₁ =15;

The unmatched harmonic interference d ₀ and the matched harmonic interference d ₁ can be described by the following external models respectively:

Among them, w and ξ are the state of the external model, and the coefficient matrix V ₀ =V ₁ =[1 0];

The unknown nonlinear function f(x ₁ ,x ₂ ) satisfies the first-order derivability condition, namely Among them, h is an unknown bounded function;

In the second step, interference observers are designed for the two types of harmonic interference, and the real-time estimation of harmonic interference is completed:

The disturbance observer designed for the unmatched harmonic disturbance d ₀ is:

in, represents the estimated value of _d0 , Represents the estimated value of the state w, v ₀ is the auxiliary state variable, L ₁ is the observer gain matrix;

The disturbance observer designed for the matched harmonic disturbance d ₁ is:

in, represents the estimated value _of d1, represents the estimated value of ξ, let the state x ₃ =f(x ₁ ,x ₂ ), and is the estimated value of state x ₃ , v ₁ is the auxiliary state variable, L ₂ is the observer gain matrix;

The third step is to design an extended state observer based on the output of the disturbance observer, and complete the estimation of the unknown nonlinear function and system state:

Taking x ₃ as the augmented state, the second-order system Σ ₀ can be written in the form of the augmented system:

Based on the output of disturbance observers Σ ₃ and Σ ₄ , the extended state observer for the augmented system Σ ₅ is designed as:

in, represent the estimated values of states x ₁ , x ₂ , and x ₃ respectively, represents the estimated value of y, Represents the estimated value of x, l ₁ , l ₂ , l ₃ represent the gain of the expanded state observer;

Combining the external model Σ ₁ and the disturbance observer Σ ₃ , the estimation error of the mismatched harmonic disturbance d ₀ can be obtained The dynamic equation of :

Combining the external model Σ ₂ with the disturbance observer Σ ₄ , the estimation error of the matched harmonic disturbance d ₁ can be obtained The dynamic equation of :

in, Indicates the estimation error of state x ₃ ;

Similarly, combining the augmented system Σ ₅ and the extended state observer Σ ₆ , the state estimation error is obtained Dynamic equations for i=1,2,3:

Combining the above three types of dynamic equations and performing corresponding transformations, we can get:

in, The specific expression of the coefficient matrix is as follows:

C ₁ =[0 0 1],

The fourth step, combined with the estimated value of non-matching interference, completes the coordinate transformation by introducing a new state variable:

Based on the estimated values of unmatched harmonic disturbance d ₀ and matched harmonic disturbance d ₁ , the second-order system Σ ₀ can be transformed into:

in,

Neglect the estimation error of interference and state and introduce a new state variable z ₁ =x ₁ , z ₃ ＝x ₃ , the following transformed control system can be obtained:

The fifth step is to design an ADRC based on the output of the disturbance observer and the extended state observer on the basis of the above coordinate transformation:

For the system Σ ₉ , the ADRC based on the disturbance observer is designed as:

Among them, p ₁ and p ₂ are controller gains,

In the sixth step, based on the separation theorem and the pole configuration theory, the gain solution of the observer and the controller is completed, thereby completing the design of the controller:

Based on the linear system separation theorem, the disturbance observer gain matrices L ₁ and L ₂ , the extended state observer gain matrix L and the controller gains p ₁ and p ₂ can be solved by pole configuration respectively:

Among them, s represents a complex variable, I represents an identity matrix of appropriate dimension, the symbol |·| represents the determinant of solving a square matrix, ω ₀ >0, ω ₁ >0 are given constants, which represent the bandwidth of the system. In this implementation case, the value of each element in L and K is calculated to be between -5 and 5, and the values of parameters p ₁ and p ₂ are between -20 and 20.

The contents not described in detail in the description of the present invention belong to the prior art known to those skilled in the art.

Claims (7)

1. A non-matching interference system active disturbance rejection control method based on a disturbance observer is characterized in that: the method comprises 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 non-matching disturbance system active disturbance rejection control method 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:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>0</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>+</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>

wherein x is₁And x₂In order to be in the state of the system,andis the time derivative of the system state, y is the measurement output, x ═ x₁x₂]^TU is the control input, b is a constant greater than zero, f (x)₁,x₂) Is a first order derivable unknown non-linear function; d₀And d₁Respectively represent the non-matching harmonic interference and the matching harmonic interference of which the frequency information is known and are characterized byD₀And D₁Which represents the magnitude of the unknown signal,andrepresenting unknown phase, ω₀And omega₁Representing a known frequency, t representing a time instant;

non-matching harmonic interference d₀And matched harmonic interference d₁Described by the following external models, respectively:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>1</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mover> <mi>w</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>W</mi> <mn>0</mn> </msub> <mi>w</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <mi>w</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>2</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mover> <mi>&amp;xi;</mi> <mo>&amp;CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>W</mi> <mn>1</mn> </msub> <mi>&amp;xi;</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mi>&amp;xi;</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>

where w and ξ are the states of the external model, the coefficient matrixV₀＝V₁＝[10]；

Unknown non-linear function f (x)₁,x₂) 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 d₀Designing a disturbance observer as follows:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>3</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <mover> <mi>w</mi> <mo>^</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>w</mi> <mo>^</mo> </mover> <mo>=</mo> <msub> <mi>v</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>W</mi> <mn>0</mn> </msub> <mover> <mi>w</mi> <mo>^</mo> </mover> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>

wherein,denotes d₀Is determined by the estimated value of (c),an estimate value, v, representing the state w₀For auxiliary state variables, L₁Is an observer gain matrix;

for matching harmonic interference d₁Designing a disturbance observer as follows:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>4</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mover> <mi>&amp;xi;</mi> <mo>^</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>&amp;xi;</mi> <mo>^</mo> </mover> <mo>=</mo> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>W</mi> <mn>1</mn> </msub> <mover> <mi>&amp;xi;</mi> <mo>^</mo> </mover> <mo>-</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mi>b</mi> <mi>u</mi> <mo>+</mo> <mi>b</mi> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>

wherein,denotes d₁Is determined by the estimated value of (c),representing an estimate of ξ, order state x₃＝f(x₁,x₂) To do soIs a state x₃Estimated value of v₁For auxiliary state variables, L₂Is the observer gain matrix.

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 be₃As an augmented state, a second order system ∑₀Writing is in the form of an augmentation system:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>5</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>d</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>+</mo> <mi>b</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>+</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <mi>h</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>

based on interference observer sigma₃And sigma₄Output of (2) to the augmentation System ∑₅The extended state observer is designed as follows:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>6</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>(</mo> <mi>y</mi> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <mi>b</mi> <mo>(</mo> <mi>u</mi> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> <mo>+</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>(</mo> <mi>y</mi> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mover> <mi>x</mi> <mo>^</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>l</mi> <mn>3</mn> </msub> <mo>(</mo> <mi>y</mi> <mo>-</mo> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>)</mo> </mtd> </mtr> <mtr> <mtd> <mover> <mi>y</mi> <mo>^</mo> </mover> <mo>=</mo> <mover> <mi>x</mi> <mo>^</mo> </mover> </mtd> </mtr> </mtable> </mfenced> </mrow>

wherein,respectively represent the state x₁,x₂,x₃Is determined by the estimated value of (c),an estimate of the value of y is represented,denotes the estimated value of x,/₁,l₂,l₃Representing the gain of the extended state observer;

incorporating an external model ∑₁And disturbance observer sigma₃Obtaining the non-matching harmonic interference d₀Estimation errorA dynamic equation of (c); incorporating an external model ∑₂And disturbance observer sigma₄Obtaining the matching harmonic interference d₁Estimation errorA dynamic equation of (c); combined augmentation System ∑₅And extended state observer Σ₆Obtaining a state estimation error1,2, 3; combining the dynamic equations to obtain:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>7</mn> </msub> <mo>:</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mover> <mi>w</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mover> <mi>&amp;xi;</mi> <mo>~</mo> </mover> <mo>&amp;CenterDot;</mo> </mover> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mo>-</mo> <mi>L</mi> <mi>C</mi> </mrow> </mtd> <mtd> <mi>M</mi> </mtd> <mtd> <mi>N</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>W</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>V</mi> <mn>0</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>W</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>bV</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mi>x</mi> <mo>~</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>w</mi> <mo>~</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>&amp;xi;</mi> <mo>~</mo> </mover> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mi>h</mi> <mo>&amp;OverBar;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow>

wherein,the specific expression of the coefficient matrix is as follows:

C₁＝[0 0 1]，

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 d₀And matched harmonic interference d₁Estimated value of (1), second order system ∑₀Conversion to:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>8</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>0</mn> </msub> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>~</mo> </mover> <mn>0</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>y</mi> <mo>=</mo> <mi>x</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>

wherein,

neglecting the estimation error of interference and state, combining the estimation value of non-matching interferenceAnd introduces a new state variable z₁＝x₁，z₃＝x₃Obtaining a control system after transformation as follows:

<mrow> <msub> <mi>&amp;Sigma;</mi> <mn>9</mn> </msub> <mo>:</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mo>+</mo> <mi>b</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>+</mo> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <msub> <mi>W</mi> <mn>0</mn> </msub> <mover> <mi>w</mi> <mo>^</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>z</mi> <mo>&amp;CenterDot;</mo> </mover> <mn>3</mn> </msub> <mo>=</mo> <mi>h</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>

6. the disturbance observer-based non-matching disturbance system active disturbance rejection control method according to claim 1, wherein: the fifth step is specifically realized as follows:

according to the control system ∑₉Designing an active disturbance rejection controller based on a disturbance observer as follows:

<mrow> <mi>u</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mi>b</mi> </mfrac> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <msub> <mover> <mi>z</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <msub> <mover> <mi>z</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mover> <mi>z</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <msub> <mi>W</mi> <mn>0</mn> </msub> <mover> <mi>w</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> </mrow>

wherein p is₁,p₂In order to control the gain of the controller,

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 theorem₁And L₂And an expanded stateGain matrix L of a detector and a controller gain p₁,p₂Solving by pole allocation respectively:

<mrow> <mrow> <mo>|</mo> <mrow> <mi>s</mi> <mi>I</mi> <mo>-</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>A</mi> <mo>-</mo> <mi>L</mi> <mi>C</mi> </mrow> </mtd> <mtd> <mi>M</mi> </mtd> <mtd> <mi>N</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>W</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <msub> <mi>V</mi> <mn>0</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>W</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <msub> <mi>bV</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>s</mi> <mo>+</mo> <msub> <mi>&amp;omega;</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mn>10</mn> </msup> </mrow>

<mrow> <mrow> <mo>|</mo> <mrow> <mi>s</mi> <mi>I</mi> <mo>-</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>p</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>p</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>s</mi> <mo>+</mo> <msub> <mi>&amp;omega;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow>

where s represents a complex variable, I represents an identity matrix of appropriate dimensions, the symbol | · | represents the determinant for solving the square matrix, ω₀>0、ω₁>0 is a given constant, representing the bandwidth of the system.