CN102749852A - Fault-tolerant anti-interference control method for multisource interference system - Google Patents

Abstract

The invention relates to a fault-tolerant anti-interference control method for a multisource interference system. Aiming at the multisource interference system containing time-varying faults, modeling interference and non-modeling random interference, a fault-tolerant anti-interference controller is designed. The fault-tolerant anti-interference control method comprises the following steps of: firstly, designing a fault diagnosis observer to estimate and counteract the time-varying faults in the system; secondly, designing an interference observer to estimate and counteract the modeling interference in the multisource interference system; thirdly, designing a robust H infinity state feedback controller to inhibit the non-modeling random interference, fault estimation errors and interference estimation errors in the multisource interference system; and finally, designing the fault-tolerant anti-interference controller based on the fault diagnosis observer, the interference observer and the robust H infinity state feedback controller. The method has the advantages of high anti-interference performance, significant fault-tolerant performance, high working reliability and the like and can be used for altitude control subsystems in the fields of aviation and aerospace.

Description

Fault-tolerant anti-interference control method of multi-source interference system

Technical Field

The invention relates to a fault-tolerant anti-interference control method, in particular to a fault-tolerant anti-interference control method of a multi-source interference system, which can be used for fault diagnosis and fault-tolerant anti-interference control of the multi-source interference system, such as an attitude control subsystem of an aerospace system of a satellite, a missile, an airplane and the like.

Background

In recent years, with the development of aerospace technology, the structural and task requirements of the aircraft are increasingly complex, the requirements on control precision and stability are higher and higher, and the failure occurrence probability of the aircraft is higher and higher. For example, through statistics and analysis of 764 spacecraft which were successfully launched at home and abroad in 1990-2001, 121 failed, which accounts for 15.8% of the total number of the spacecraft, some students found. The reliability of aircraft operation, the maintainability and the effectiveness of on-orbit flight have become a major research focus in the field of aerospace. Fault-tolerant control and fault detection and diagnosis open up a new way for improving the reliability, maintainability and effectiveness of the system. In addition, with the increasing complexity of the aircraft structure and task requirements, more and more factors influencing the attitude control precision and stability of the aircraft are mainly summarized as the following points: external environment disturbance moment, vibration of a flywheel of an actuating mechanism in the star body, friction of the flywheel, air injection momentum unloading, sensor measurement noise, uncertainty of a system model and the like.

Aiming at the series of problems, when time-varying energy bounded faults exist in the system but no interference exists, domestic and foreign scholars put forward a plurality of effective methods and obtain certain effects. However, considering that the interference is not present and is not present in the actual system, a single fault consideration may cause a series of problems, and particularly, the attitude control accuracy and stability of the system cannot be guaranteed. On the basis, the situation that time-varying faults and interference exist in the system at the same time becomes a research direction, and domestic and foreign scholars propose a series of solutions, such as H_∞Optimization technology and H based on internal mold structure_∞A controller, etc. However, the existing method considers the interference as norm bounded quantity to suppress on the basis of designing the observer to estimate the fault, and such a processing method has the following disadvantages: first of allAll interferences in the system are regarded as norm bounded quantities to be inhibited, namely the interferences existing in the system are regarded as a whole to be processed, the information of the intra-system part modeling interferences is ignored or the interference information which can be obtained by a physical measurement means is not fully utilized, the system resources are not fully utilized, and the high-precision attitude control is difficult to realize; secondly, all the interferences in the system are regarded as norm bounding quantities to be suppressed, so that the conservatism of the system is bound to be increased, and the high-precision attitude control is difficult to realize.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: aiming at a multi-source interference system, the defects of the prior art are overcome, the fault-tolerant anti-interference control method with interference cancellation and inhibition performance is provided, the problems of interference cancellation, interference inhibition, fault diagnosis and fault-tolerant control of the multi-source interference system are solved, and the control precision and stability of the system are improved.

The technical solution of the invention is as follows: a fault-tolerant anti-interference control method of a multi-source interference system is characterized by comprising the following steps:

firstly, designing a fault diagnosis observer to estimate and counteract time-varying faults in a system; secondly, designing a disturbance observer to estimate and counteract modelable disturbance in the multi-source disturbance system; thirdly, design robust H_∞The state feedback controller inhibits unmoldable random interference, fault estimation errors and interference estimation errors in the multi-source interference system; finally, based on fault diagnosis observer, disturbance observer and robust H_∞A state feedback controller, which is designed as a fault-tolerant anti-interference controller; the method comprises the following specific steps:

firstly, building a dynamic model containing a multi-source interference system, and writing a state space expression

Aiming at a multi-source interference system containing time-varying faults, modelable interference and unmodeled random interference, a system dynamics model is built and written into a state space expression as follows:

<math> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

wherein x (t) is the state variable of the multi-source interference system, u (t) is the control input, d₁(t) modelable interference, F (t) time-varying fault, d₂(t) unmoldable random interference, A, E, B₁And B₂Is a matrix of known dimensions and is,

for system nonlinear terms and satisfying the Lipschitz condition, the external model describes the disturbance d₁(t) from the following external interference model ∑₁Represents:

<math> <mrow> <msub> <mi>&Sigma;</mi> <mn>1</mn> </msub> <mo>:</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Vw</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>w</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Ww</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>3</mn> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein W (t) is a state variable of the modelable interference model, V is an output matrix of the modelable interference model, W represents a system matrix of the modelable interference model, delta (t) is the non-modelable random interference with bounded energy, and B₃A gain array of random interference is unmoldable.

Second, designing a fault diagnosis observer

Aiming at the time-varying fault F (t) in the multi-source interference system, a fault diagnosis observer is designed to estimate the time-varying fault F (t) in real time, and an estimated value is obtained

Further obtainError of fault estimation

Thirdly, designing a disturbance observer

Modeling disturbance d in multi-source disturbance system₁(t) designing a disturbance observer to estimate the disturbance observer in real time and obtaining an estimated value

Further obtaining interference estimation error Is an estimate of w (t).

The fourth step, design robust H_∞State feedback controller

For unmoldable random disturbance d in multi-source disturbance system₂(t) error of failure estimation e_F(t) and interference estimation error e_w(t), designing robust H_∞The state feedback controller restrains the state feedback, and the controller structure is as follows:

u_f(t)＝Mx(t)

wherein u is_f(t) is robust H_∞And (4) inputting state feedback control, wherein M is a gain array of the feedback controller in an undetermined state.

Fifthly, designing a fault-tolerant anti-interference controller

Designing fault-tolerant anti-interference controller to time-varying fault F (t) and modelable interference d in system₁(t) counteracting, unmodeled random interference d₂(t) error of failure estimation e_F(t) and interference estimation error e_w(t) suppressing, wherein the fault-tolerant anti-interference controller has the following structure:

$u\left(t\right)=u_f\left(t\right)-\hat{F}\left(t\right)-{\hat{d}}_1\left(t\right)$

the multi-source interference system can be expressed as:

<math> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>M</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>V</mi> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

the systematic estimation error equation for organizing the modelable interference model and the systematic estimation error equation for the time-varying fault are as follows:

<math> <mrow> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>LB</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>3</mn> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

<math> <mrow> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mi>V</mi> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>KB</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

combining the multi-source interference system, the system estimation error equation of the time-varying fault and the system estimation error equation of the modeling interference to obtain a closed-loop system:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>A</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>M</mi> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>LB</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>KB</mi> <mn>1</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>LB</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>KB</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mo>&infin;</mo> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>C</mi> <mn>0</mn> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mn>1</mn> </msub> </mtd> <mtd> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> </mfenced> </math>

wherein z is_∞(t) is H_∞Performance reference output, [ C ]₀C₁C₂]Is H_∞And (5) outputting a matrix with adjustable performance.

Sixth, gain matrix solution

Solving a fault-tolerant anti-interference controller gain array of the multi-source interference system by using a convex optimization algorithm; initial values x (0), e are given_w(0) And e_F(0) Adjustable output matrix [ C ]₀C₁C₂]The non-linear weight parameter lambda, the interference suppression degree gamma₁、γ₂And gamma₃Solving the following convex optimization problem:

$\min \left(\begin{array}{ccc}\left(\begin{array}{cc}{x}^T\left(0\right)& e^T\left(0\right)\end{array}\right)& \left(\begin{array}{cc}{P}_1& \\ & P_2\end{array}\right)& {\left(\begin{array}{cc}{x}^T\left(0\right)& e^T\left(0\right)\end{array}\right)}^T\end{array}\right)$

<math> <mrow> <mi>&Phi;</mi> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>&Phi;</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>G</mi> </mtd> <mtd> <mo>-</mo> <mi>E</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&Phi;</mi> <mn>18</mn> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mn>1</mn> </msub> <msubsup> <mi>C</mi> <mn>0</mn> <mi>T</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mi>&Phi;</mi> <mn>22</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>P</mi> <mn>2</mn> </msub> <msub> <mi>H</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>R</mi> <mn>2</mn> </msub> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mn>2</mn> </msub> <msub> <mi>H</mi> <mn>2</mn> </msub> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <mi>U</mi> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>G</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msup> <mi>&lambda;</mi> <mn>2</mn> </msup> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <mi>UE</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>UB</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein e (0) ═ e_w(0)e_F(0)]^T,Φ₁₁＝(AP₁+B₁R₁)+(AP₁+B₁R₁)^T,Φ₂₂＝(P₂W₁+R₂B₁G)+(P₂W₁+R₂B₁G)^T,Φ₁₈＝λ(AP₁+B₁R₁)^T,C＝[C₁C₂],G＝[EI],H₁＝[B₃ 0]^T，H₂＝[0 1]^T(ii) a Symbol represents the symmetric block of the corresponding part in the symmetric matrix, and P is obtained by solving₁、P₂、R₁And R₂The gain array of the disturbance observer and the fault diagnosis observer is $\left[\begin{array}{c}L\\ K\end{array}\right]=P_2^{-1}{R}_2,$ The state feedback controller gain array is

The structure of the fault diagnosis observer in the step 2 is as follows:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mover> <mi>F</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&tau;</mi> <mo>-</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>&tau;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>&tau;</mi> <mo>-</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mo>[</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>]</mo> </mtd> </mtr> </mtable> </mfenced> </math>

wherein,k is a gain matrix of the fault diagnosis observer to be determined, and epsilon (t) is an auxiliary variable.

The structure of the disturbance observer in the step 3 is as follows:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>V</mi> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Lx</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>v</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Lx</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>L</mi> <mo>[</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>]</mo> </mtd> </mtr> </mtable> </mfenced> </math>

wherein, V (t) is an auxiliary variable, L is a gain matrix of the interference observer to be determined, V is an output matrix of the modelable interference model, and W represents a system array of the modelable interference model.

Compared with the prior art, the invention has the advantages that:

(1) the invention relates to a fault-tolerant anti-interference control method of a multi-source interference system, which is a composite layered anti-interference control method.A feedforward part of a controller consists of a fault diagnosis observer and an interference observer and is used for estimating and offsetting time-varying faults and modelable interference in the system, and a feedback part of the controller consists of a robust H_∞The state feedback controller is formed and designedThe controller enables the system to have more fine fault diagnosis and fault-tolerant control capability.

(2) The method has strong robustness on interference, and under the condition that multi-source interference such as time-varying fault with bounded change rate, modelable interference, unmoldable random interference and the like exists at the same time, the fault diagnosis observer in the method can estimate and cancel the time-varying fault, the interference observer can estimate and cancel the modelable interference, and the robustness H is high_∞The state feedback controller inhibits unmoldable random interference, fault estimation errors and interference estimation errors, and solves the problem of high conservation caused by the existing method that the interference is regarded as norm bounded quantity to be inhibited.

Drawings

Fig. 1 is a design flow chart of a fault-tolerant anti-interference control method based on a multi-source interference system according to the present invention.

Detailed Description

As shown in fig. 1, the implementation steps of the present invention are as follows (the satellite attitude determination and control system is taken as an example to illustrate the implementation of the method):

1. building a dynamic model containing a multi-source interference system and writing a state space expression

When the Euler angle between the micro-nano satellite body coordinate system and the orbit coordinate system is small, the following satellite linear attitude dynamics and kinematics models can be obtained:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>J</mi> <mn>1</mn> </msub> <mover> <mi>&phi;</mi> <mrow> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> </mrow> </mover> <mo>-</mo> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>J</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mover> <mi>&psi;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <mn>4</mn> <msup> <mi>n</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>J</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>&phi;</mi> <mo>=</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mn>2</mn> </msub> <mover> <mi>&theta;</mi> <mrow> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> </mrow> </mover> <mo>+</mo> <mn>3</mn> <msup> <mi>n</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>J</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mi>&theta;</mi> <mo>=</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mn>3</mn> </msub> <mover> <mi>&psi;</mi> <mrow> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> </mrow> </mover> <mo>+</mo> <mi>n</mi> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>J</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>J</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mi>&psi;</mi> <mo>=</mo> <msub> <mi>u</mi> <mn>3</mn> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow> <mi>d</mi> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> </math>

in the above formula, J₁,J₂,J₃The rotational inertia is three axes, n is the satellite orbit angular velocity, phi, theta and psi are three axes Euler angles between the satellite body coordinate system and the orbit coordinate system;

respectively, the three-axis euler angular rates;

respectively, three-axis euler angular acceleration; u. of₁,u₂,u₃Three-axis control moment respectively; f (T) is a time-varying fault, T_d1,T_d2,T_d3Disturbance moments (including disturbance moments due to sensor or actuator failure) of three axes respectively;

uncertainty of the micro-nano satellite model mainly comes from uncertainty of rotational inertia, an inertia matrix is extracted from the attitude dynamics model by considering the uncertainty of the rotational inertia, and the above formula can be converted into the following form:

<math> <mrow> <mrow> <mo>(</mo> <mi>M</mi> <mo>+</mo> <mi>&Delta;M</mi> <mo>)</mo> </mrow> <mover> <mi>p</mi> <mrow> <mo>&CenterDot;</mo> <mo>&CenterDot;</mo> </mrow> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mi>C</mi> <mo>+</mo> <mi>&Delta;C</mi> <mo>)</mo> </mrow> <mover> <mi>p</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mi>S</mi> <mo>+</mo> <mi>&Delta;S</mi> <mo>)</mo> </mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>B</mi> <mi>u</mi> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mi>w</mi> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

wherein the state variable p (t) is [ phi, theta, psi]^TIs a three-axis Euler angle,

is the three-axis euler angular velocity,

is the three-axis Euler angular acceleration, d₁(t) modelable interference, d₂(t) is an energy bounded, unmodeled random (i.e., L)₂Norm of

Bounded) interference, B_uAssigning a matrix to the control inputs, B_wA matrix is allocated to unmodeled random disturbance inputs, M, C, S are known moments of inertia, Δ M, Δ C, Δ S are uncertain moments of inertia due to disturbances, $M=\left[\begin{array}{ccc}{I}_1& 0& 0\\ 0& I_2& 0\\ 0& 0& I_3\end{array}\right],$ <math> <mrow> <mi>C</mi> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>-</mo> <mi>&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>&omega;</mi> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>+</mo> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow> </math> <math> <mrow> <mi>S</mi> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>4</mn> <msup> <mi>&omega;</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <msup> <mi>&omega;</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>I</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>-</mo> <msup> <mi>&omega;</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>I</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </math> $B_u=B_w=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],$ the above formula is arranged and converted into a state space model as shown in the following formula:

<math> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

wherein the state variables of the multisource interference system <math> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <mi>t</mi> </msubsup> <msub> <mi>e</mi> <mi>q</mi> </msub> <mrow> <mo>(</mo> <mi>&tau;</mi> <mo>)</mo> </mrow> <mi>d&tau;</mi> </mtd> <mtd> <msub> <mi>e</mi> <mi>q</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> <mtd> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>q</mi> </msub> </mtd> </mtr> </mtable> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mfenced> <mi>T</mi> </msup> <mo>,</mo> </mrow> </math> e_q(t)＝p(t)-p_p(t)，p_p(t) is ginsengTaking the track signal u (t) as control input, A, E, B₁And B₂Is a matrix of known dimensions and is, $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& -M^{-1}S& -M^{-1}C\end{array}\right],$ <math> <mrow> <mi>E</mi> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>M</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>&Delta;S</mi> </mtd> <mtd> <msup> <mi>M</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>&Delta;C</mi> </mtd> <mtd> <msup> <mi>M</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>&Delta;M</mi> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> </mrow> </math> $B_1=\left[\begin{array}{c}0\\ 0\\ M^{-1}{B}_u\end{array}\right],$ $B_2=\left[\begin{array}{c}0\\ 0\\ M^{-1}{B}_w\end{array}\right],$ the nonlinear term f: (

(t)) meets the Lipschitz condition, namely that the known Lipschitz parameter array U belongs to R^3×3Such that the following inequality holds:

<math> <mrow> <mo>|</mo> <mo>|</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> <mo>&le;</mo> <mo>|</mo> <mo>|</mo> <mi>U</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> </mrow> </math>

wherein,for any two states in the system state set, the external model describes the disturbance d₁(t) from the following external interference model ∑₁Represents:

<math> <mrow> <msub> <mi>&Sigma;</mi> <mn>1</mn> </msub> <mo>:</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Vw</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>w</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Ww</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>3</mn> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein W (t) is a state variable of the modelable interference model, V is an output matrix of the modelable interference model, W represents a system matrix of the modelable interference model, and δ (t) is an energy-bounded unmoldable random (i.e., L)₂Norm of

Bounded) interference, B₃An energy-bounded interference gain array for the unmoldable interference model.

2. Design fault diagnosis observer

Aiming at a time-varying fault F (t) in a multi-source interference system, a fault diagnosis observer is designed as follows:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mover> <mi>F</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&tau;</mi> <mo>-</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>&tau;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>&tau;</mi> <mo>-</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mo>[</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>]</mo> </mtd> </mtr> </mtable> </mfenced> </math>

wherein ε (t) is an auxiliary variable,k is a gain matrix of the fault diagnosis observer to be determined, and the gain matrix is obtained through the subsequent step 6, so that a fault estimation error is obtained

3. Design disturbance observer

Modeling interference d in multi-source interference system₁(t), designing a disturbance observer as follows:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>V</mi> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Lx</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>v</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Lx</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>L</mi> <mo>[</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>]</mo> </mtd> </mtr> </mtable> </mfenced> </math>

wherein,for modellable disturbance d₁(ii) an estimate of the value of (t),

is the estimated value of w (t), v (t) is an auxiliary variable, L is a gain matrix of the disturbance observer to be determined, and the estimated error of the fault is obtained through the subsequent step 6 $e_w\left(t\right)=w\left(t\right)-\hat{w}\left(t\right).$

4. Design robust H_∞State feedback controller

For unmoldable random disturbance d in multi-source disturbance system₂(t) error of failure estimation e_F(t) and interference estimation error e_w(t) design robustness H_∞The state feedback controller restrains the state feedback, and the controller structure is as follows:

u_f(t)＝Mx(t)

wherein u is_fAnd (t) is a state feedback controller, and M is a gain array of the feedback controller to be determined.

5. Designing fault-tolerant anti-interference controller

Based on interference observer, fault diagnosis observer and robust H_∞The state feedback controller is designed as follows：

$u\left(t\right)=u_f\left(t\right)-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right)$

The multi-source interference system can be expressed as:

<math> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>M</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>V</mi> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

the systematic estimation error equation for organizing the modelable disturbance model and the systematic estimation error equation for the rate-of-change bounded time-varying fault are as follows:

<math> <mrow> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>LB</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>3</mn> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

<math> <mrow> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mi>V</mi> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>KB</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

combining the multi-source interference system, the system estimation error of the modeling interference model and the system estimation error equation of the time-varying fault to obtain a closed-loop system:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>A</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>M</mi> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>LB</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>KB</mi> <mn>1</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>LB</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>KB</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mo>&infin;</mo> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>C</mi> <mn>0</mn> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mn>1</mn> </msub> </mtd> <mtd> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> </mfenced> </math>

wherein z is_∞(t) is H_∞Performance reference output, [ C ]₀C₁C₂]Is H_∞And (5) outputting a matrix with adjustable performance.

6. Gain matrix solving

Solving a fault-tolerant anti-interference controller gain array of the multi-source interference system by using a convex optimization algorithm; initial values x (0), e are given_w(0) And e_F(0) Adjustable output matrix [ C ]₀C₁C₂]The non-linear weight parameter lambda, the interference suppression degree gamma₁、γ₂And gamma₃Solving the following convex optimization problem:

$\min \left(\begin{array}{ccc}\left(\begin{array}{cc}{x}^T\left(0\right)& e^T\left(0\right)\end{array}\right)& \left(\begin{array}{cc}{P}_1& \\ & P_2\end{array}\right)& {\left(\begin{array}{cc}{x}^T\left(0\right)& e^T\left(0\right)\end{array}\right)}^T\end{array}\right)$

<math> <mrow> <mi>&Phi;</mi> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>&Phi;</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>G</mi> </mtd> <mtd> <mo>-</mo> <mi>E</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&Phi;</mi> <mn>18</mn> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mn>1</mn> </msub> <msubsup> <mi>C</mi> <mn>0</mn> <mi>T</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mi>&Phi;</mi> <mn>22</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>P</mi> <mn>2</mn> </msub> <msub> <mi>H</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>R</mi> <mn>2</mn> </msub> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mn>2</mn> </msub> <msub> <mi>H</mi> <mn>2</mn> </msub> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <mi>U</mi> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>G</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msup> <mi>&lambda;</mi> <mn>2</mn> </msup> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <mi>UE</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>UB</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein e (0) ═ e_w(0)e_F(0)]^T,Φ₁₁＝(AP₁+B₁R₁)+(AP₁+B₁R₁)^T,Φ₂₂＝(P₂W₁+R₂B₁G)+(P₂W₁+R₂B₁G)^T,Φ₁₈＝λ(AP₁+B₁R₁)^T,C＝[C₁C₂],G＝[E I],H₁＝[B₃ 0]^T,H₂＝[0 1]^T(ii) a Symbol represents the symmetric block of the corresponding part in the symmetric matrix, and P is obtained by solving₁、P₂、R₁And R₂The gain array of the disturbance observer and the fault diagnosis observer is $\left[\begin{array}{c}L\\ K\end{array}\right]=P_2^{-1}{R}_2,$ The state feedback controller gain array is

Those skilled in the art will appreciate that the invention may be practiced without these specific details.

Claims (3)

1. A fault-tolerant anti-interference control method of a multi-source interference system is characterized by comprising the following steps: firstly, designing a fault diagnosis observer to estimate and counteract time-varying faults in a system; secondly, designing a disturbance observer to estimate and counteract modelable disturbance in the multi-source disturbance system; thirdly, design robust H_∞The state feedback controller inhibits unmoldable random interference, fault estimation errors and interference estimation errors in the multi-source interference system; finally, based on fault diagnosis observer, disturbance observer and robust H_∞State feedback controller, design fault-tolerant anti-jammingA controller; the method comprises the following specific steps:

firstly, building a dynamic model containing a multi-source interference system, and writing a state space expression

Aiming at a multi-source interference system containing time-varying faults, modelable interference and unmodeled random interference, a system dynamics model is built and written into a state space expression as follows:

<math> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>F</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

wherein x (t) is the state variable of the multi-source interference system, u (t) is the control input, d₁(t) modelable interference, F (t) time-varying fault, d₂(t) unmoldable random interference, A, E, B₁And B₂Is a matrix of known dimensions and is,for system nonlinear terms and satisfying the Lipschitz condition, the external model describes the disturbance d₁(t) from the following external interference model ∑₁Represents:

<math> <mrow> <msub> <mi>&Sigma;</mi> <mn>1</mn> </msub> <mo>:</mo> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Vw</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>w</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>Ww</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>3</mn> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein W (t) is a state variable of the modelable interference model, V is an output matrix of the modelable interference model, W represents a system matrix of the modelable interference model, delta (t) is the non-modelable random interference with bounded energy, and B₃A gain array for unmoldable random interference;

second, designing a fault diagnosis observer

For time varying in multi-source jamming systemsAnd designing a fault diagnosis observer to estimate the fault in real time and obtaining an estimated value

Further obtain the fault estimation error

Thirdly, designing a disturbance observer

Modeling disturbance d in multi-source disturbance system₁(t) designing a disturbance observer to estimate the disturbance observer in real time and obtaining an estimated valueFurther obtaining interference estimation error

Is an estimate of w (t);

the fourth step, design robust H_∞State feedback controller

For unmoldable random disturbance d in multi-source disturbance system₂(t) error of failure estimation e_F(t) and interference estimation error e_w(t), designing robust H_∞The state feedback controller restrains the state feedback, and the controller structure is as follows:

u_f(t)＝Mx(t)

wherein u is_f(t) is robust H_∞A state feedback control input, wherein M is a gain array of a feedback controller in an undetermined state;

fifthly, designing a fault-tolerant anti-interference controller

Designing fault-tolerant anti-interference controller to time-varying fault F (t) and modelable interference d in system₁(t) counteracting, unmodeled random interference d₂(t) error of failure estimation e_F(t) and interference estimation error e_w(t) suppression, fault tolerant anti-jamming controlThe structure of the device is as follows:

$u\left(t\right)=u_f\left(t\right)-\hat{F}\left(t\right)-{\hat{d}}_1\left(t\right)$

the multi-source interference system can be expressed as:

<math> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>A</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>M</mi> <mo>)</mo> </mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>V</mi> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

the systematic estimation error equation for organizing the modelable interference model and the systematic estimation error equation for the time-varying fault are as follows:

<math> <mrow> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> <mo>)</mo> </mrow> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>LB</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>3</mn> </msub> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

<math> <mrow> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mi>V</mi> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>KB</mi> <mn>2</mn> </msub> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>

combining the multi-source interference system, the system estimation error equation of the time-varying fault and the system estimation error equation of the modeling interference to obtain a closed-loop system:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>A</mi> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>M</mi> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>LB</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mi>V</mi> </mtd> <mtd> <msub> <mi>KB</mi> <mn>1</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>LB</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>KB</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>B</mi> <mn>3</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>&delta;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mover> <mi>F</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>f</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>z</mi> <mo>&infin;</mo> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>C</mi> <mn>0</mn> </msub> </mtd> <mtd> <msub> <mi>C</mi> <mn>1</mn> </msub> </mtd> <mtd> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open='(' close=')'> <mtable> <mtr> <mtd> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>w</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mfenced> </mtd> </mtr> </mtable> </mfenced> </math>

wherein z is_∞(t) is H_∞Performance reference output, [ C ]₀C₁C₂]Is H_∞A performance adjustable output matrix;

sixth, gain matrix solution

Solving a fault-tolerant anti-interference controller gain array of the multi-source interference system by using a convex optimization algorithm; initial values x (0), e are given_w(0) And e_F(0) Adjustable output matrix [ C ]₀C₁C₂]The non-linear weight parameter lambda, the interference suppression degree gamma₁、γ₂And gamma₃Solving the following convex optimization problem:

$\min \left(\begin{array}{ccc}\left(\begin{array}{cc}{x}^T\left(0\right)& e^T\left(0\right)\end{array}\right)& \left(\begin{array}{cc}{P}_1& \\ & P_2\end{array}\right)& {\left(\begin{array}{cc}{x}^T\left(0\right)& e^T\left(0\right)\end{array}\right)}^T\end{array}\right)$

<math> <mrow> <mi>&Phi;</mi> <mo>=</mo> <mfenced open='[' close=']'> <mtable> <mtr> <mtd> <msub> <mi>&Phi;</mi> <mn>11</mn> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>G</mi> </mtd> <mtd> <mo>-</mo> <mi>E</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>&Phi;</mi> <mn>18</mn> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mn>1</mn> </msub> <msubsup> <mi>C</mi> <mn>0</mn> <mi>T</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <msub> <mi>&Phi;</mi> <mn>22</mn> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>P</mi> <mn>2</mn> </msub> <msub> <mi>H</mi> <mn>1</mn> </msub> </mtd> <mtd> <msub> <mi>R</mi> <mn>2</mn> </msub> <msub> <mi>B</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>P</mi> <mn>2</mn> </msub> <msub> <mi>H</mi> <mn>2</mn> </msub> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <mi>U</mi> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>G</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <msup> <mi>C</mi> <mi>T</mi> </msup> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msup> <mi>&lambda;</mi> <mn>2</mn> </msup> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <mi>UE</mi> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>&lambda;</mi> <msup> <mrow> <mo>(</mo> <msub> <mi>UB</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <msubsup> <mi>&gamma;</mi> <mn>3</mn> <mn>2</mn> </msubsup> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <mi>I</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>*</mo> </mtd> <mtd> <mo>-</mo> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> </mrow> </math>

wherein e (0) ═ e_w(0)e_F(0)]^T,Φ₁₁＝(AP₁+B₁R₁)+(AP₁+B₁R₁)^T,Φ₂₂＝(P₂W₁+R₂B₁G)+(P₂W₁+R₂B₁G)^T,Φ₁₈＝λ(AP₁+B₁R₁)^T,C＝[C₁ C₂],G＝[E I],H₁＝[B₃ 0]^T,H₂＝[0 1]^T(ii) a Symbol represents the symmetric block of the corresponding part in the symmetric matrix, and P is obtained by solving₁、P₂、R₁And R₂The gain array of the disturbance observer and the fault diagnosis observer is $\left[\begin{array}{c}L\\ K\end{array}\right]=P_2^{-1}{R}_2,$ The state feedback controller gain array is

2. The fault-tolerant anti-interference control method of the multi-source interference system according to claim 1, characterized in that: the structure of the fault diagnosis observer in the step 2 is as follows:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <mover> <mi>F</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&tau;</mi> <mo>-</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>&tau;</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>KB</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>&tau;</mi> <mo>-</mo> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mo>[</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>]</mo> </mtd> </mtr> </mtable> </mfenced> </math>

wherein,

k is a gain matrix of the fault diagnosis observer to be determined, and epsilon (t) is an auxiliary variable.

3. The fault-tolerant anti-interference control method of the multi-source interference system according to claim 1, characterized in that: the structure of the disturbance observer in the step 3 is as follows:

<math> <mfenced open='{' close=''> <mtable> <mtr> <mtd> <msub> <mover> <mi>d</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>V</mi> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>w</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Lx</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi>v</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mi>W</mi> <mo>+</mo> <msub> <mi>LB</mi> <mn>1</mn> </msub> <mi>V</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>v</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Lx</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>L</mi> <mo>[</mo> <mi>Ax</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mi>u</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mover> <mi>F</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mi>Ef</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>]</mo> </mtd> </mtr> </mtable> </mfenced> </math>

wherein, V (t) is an auxiliary variable, L is a gain matrix of the interference observer to be determined, V is an output matrix of the modelable interference model, and W represents a system array of the modelable interference model.

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