CN103116357B - A kind of sliding-mode control with anti-interference fault freedom - Google Patents

Abstract

There is a sliding-mode control for anti-interference fault freedom, for the system that contains fault and interference, design a kind of sliding mode controller with anti-interference fault freedom; First, the fault in taking into account system and multi-source disturb, and set up the kinetic model of system; Secondly, the fault in design error failure diagnostic observations device and interference observer estimating system and can modeling disturbing; Again, solve the gain matrix of interference observer and fault diagnosis observer; Then, design sliding mode controller, the estimated value of operational failure and interference is compensate for failed and interference respectively; Finally, analyzer-controller stability, inputs definite sliding formwork yield value under saturated prerequisite in system; This method has ensured the anti-interference and fault freedom of system, and there is robustness for can not modeling disturbing in system, be applicable to the multi-source EVAC of input-bound, improved the chattering phenomenon of sliding formwork control, can be used in the attitude control system in Aeronautics and Astronautics and survey of deep space field.

Description

Sliding mode control method with anti-interference fault-tolerant performance

Technical Field

The invention relates to a sliding mode control method with anti-interference fault-tolerant performance, which can be used for anti-interference fault-tolerant control of an input-limited system, such as an attitude control subsystem in the aerospace and deep space detection fields of satellites, airplanes and the like with input torque limited by the maximum rotating speed of a flywheel.

Background

With the complication of the task of the spacecraft, the requirement on the attitude control precision is higher and higher, and the high-precision attitude control of the spacecraft becomes a research hotspot at home and abroad. The space environment of the spacecraft is complex, the spacecraft is interfered by external and internal multi-source and unmodeled dynamically, and the fault occurrence probability of the attitude control system is high. The faults of the sensors, the flywheel and the like can cause task interruption and even failure, and in order to improve the reliability of the system, an accurate fault diagnosis and fault-tolerant control method is required to be adopted. Meanwhile, the aircraft attitude control system is not accurately modeled due to various factors such as unmodeled dynamics, unknown parameters, random interference and other equivalent interference variables, so that the control accuracy is reduced and even instability is caused, and the anti-interference attitude control method for the spacecraft is very important. In addition, the flywheel is limited by the maximum rotating speed, the generated input torque is limited, and the system has the problem of input saturation. Input saturation affects the control performance of the system, is prone to cause system instability, and must be considered in the design process of the controller.

In order to solve the problems, scholars at home and abroad propose a plurality of effective methods. In the case where there is a time-varying fault in which only a derivative is bounded, the fault diagnosis method is classified into a dynamic mathematical model-based method, a signal processing-based method, and a knowledge-based method, among which an observer-based method, a neural network-based method, and a study of wavelet transformation are very extensive. Under the condition that only modelable interference exists in the system, the interference can be estimated and counteracted based on the control of the interference observer, and the system has the advantages of simple structure and capability of combining different control methods according to different performance requirements of the system. None of the above methods can be applied to system control where there are limits on faults, disturbances and input saturation.

In recent years, sliding mode control receives more and more attention due to the excellent characteristics of the sliding mode control, and the method is insensitive to parameter change and disturbance, has a simple structure and is suitable for control of a satellite attitude control system. Due to the influence of faults and interference in the attitude control system, the sliding mode control method is easy to generate buffeting, so that a learner designs an interference observer to counteract the influence of the interference, the buffeting problem of the sliding mode control is improved, high-reliability and high-precision attitude control is realized, and the faults are not considered.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: aiming at an input saturated multi-source interference system, a sliding mode control method with fault compensation and interference cancellation and inhibition performance is provided. The fault diagnosis observer and the disturbance observer are designed to estimate and counteract the fault and the disturbance in the system, and the sliding mode controller is designed to have robustness to parameter uncertainty and disturbance.

The technical solution of the invention is as follows: a sliding mode control method with anti-interference fault-tolerant performance is characterized by comprising the following steps:

firstly, considering faults and multi-source interference in a system, and establishing a dynamic model of the system; secondly, designing a fault diagnosis observer and a disturbance observer to estimate faults and modelable disturbance in a system; thirdly, solving a gain matrix of the interference observer and the fault diagnosis observer; then designing a sliding mode controller, substituting the interference and fault estimation values into the controller to compensate equivalent interference and fault; finally, analyzing the stability of the controller, and solving the sliding mode gain on the premise of system input saturation; the method comprises the following specific steps:

firstly, considering faults and multi-source interference in a system, and establishing a dynamic model of the system

And (3) building a system dynamics model containing faults and interferences, as follows:

$\left\{\begin{array}{c}\stackrel{.}{x_1}\left(t\right)=x_2\left(t\right)\\ \stackrel{.}{x_2}\left(t\right)=x_3\left(t\right)\\ .\\ .\\ .\\ \stackrel{.}{x_n}\left(t\right)=-a_0{x}_n\left(t\right)-\·\·\·\·\·\·-a_{n-1}{x}_1\left(t\right)+b_1(u\left(t\right)+d_1\left(t\right)+F\left(t\right))+b_2{d}_2\left(t\right)\end{array}\right.$

wherein x is₁(t)，x₂(t)，…，x_n(t) is the system state, n ≧ 2 is a positive integer, u (t) is the control input, F (t) is the rate-of-change bounded time-varying fault, d₁(t) modelable interference, d₂(t) is unmoldable random interference. a is₀、a₁、…a_n-1And b₁、b₂Are all system internal parameters. d₁(t) may be represented by the following interference model ∑₁Represents:

${\mathrm{\Σ}}_1:\left\{\begin{array}{c}{d}_1\left(t\right)=\mathrm{Vw}\left(t\right)\\ \stackrel{\·}{w}\left(t\right)=\mathrm{Ww}\left(t\right)+B_3\mathrm{\δ}\left(t\right)\end{array}\right.$

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, B₃A gain array for unmoldable random interference, (t) an energy bounded unmoldable random interference.

Selecting a state variable X (t) ═ x₁(t)x₂(t)......x_n(t)]^TWritten as a state space expression is as follows:

$\stackrel{\·}{X}\left(t\right)=\mathrm{AX}\left(t\right)+B_1(u\left(t\right)+d_1\left(t\right)+F\left(t\right))+B_2{d}_2\left(t\right)$

wherein X (t) is system state variable, A is system array, B₁As an input matrix, B₂A gain array of random interference is unmoldable.

$B_1={\left[\begin{array}{c}0\\ 0\\ 0\\ \·\\ \·\\ \·\\ b_1\end{array}\right]}_{n\×1},$ $B_2={\left[\begin{array}{c}0\\ 0\\ 0\\ .\\ .\\ .\\ b_2\end{array}\right]}_{n\×1}$

Secondly, designing a fault diagnosis observer and a disturbance observer to respectively estimate the fault and the modelable disturbance aiming at the time-varying fault F (t) in the system, wherein the fault diagnosis observer is designed as follows:

$\left\{\begin{array}{c}\hat{F}\left(t\right)=\mathrm{\ξ}\left(t\right)-\mathrm{KX}\left(t\right))\\ \stackrel{.}{\mathrm{\ξ}}\left(t\right)=K{B}_1(\mathrm{\ξ}\left(t\right)-\mathrm{KX}\left(t\right))+K[\mathrm{AX}\left(t\right)+B_{1}u\left(t\right)+B_1\widehat{d_1}\left(t\right))]\end{array}\right.$

for modelable interference d in a system₁(t) designing a disturbance observer as follows:

$\left\{\begin{array}{c}\widehat{d_1}\left(t\right)=V\hat{w}\left(t\right)\\ \hat{w}\left(t\right)=r\left(t\right)-\mathrm{LX}\left(t\right)\\ \stackrel{.}{r}\left(t\right)=(W+L{B}_{1}V)(r\left(t\right)-\mathrm{LX}\left(t\right))+L[\mathrm{AX}\left(t\right)+B_{1}u\left(t\right)+B_1\hat{F}\left(t\right)]\end{array}\right.$

wherein,in order to be an estimate of the fault,in order to be an interference estimate,and (d) an estimated value of w (t), ξ (t) and r (t) are auxiliary variables in the fault diagnosis observer and the disturbance observer respectively, and K and L are a gain matrix of the fault diagnosis observer to be determined and a gain matrix of the disturbance observer respectively, and are obtained by the subsequent step 3.

Defining fault estimation errors $e_F\left(t\right)=F\left(t\right)-\hat{F}\left(t\right),$ Interference observation error $e_w\left(t\right)=w\left(t\right)-\hat{w}\left(t\right);$

The fault estimation error equation can be obtained according to the expression of the fault diagnosis observer as follows:

${\stackrel{.}{e}}_F\left(t\right)=K{B}_1{e}_F\left(t\right)+K{B}_{1}V{e}_w\left(t\right)+K{B}_2{d}_2\left(t\right)+\stackrel{.}{F}\left(t\right)$

the interference estimation error equation can be obtained according to the expression of the interference observer as follows:

${\stackrel{.}{e}}_w\left(t\right)=L{B}_{1}V{e}_w\left(t\right)+L{B}_1{e}_F\left(t\right)+L{B}_2{d}_2\left(t\right)+B_3\mathrm{\δ}\left(t\right)$

thirdly, solving the gain matrix of the fault diagnosis observer and the gain matrix of the interference observer

The estimation error equation for modelable interference and the estimation error equation for fault in the second step of the cascade are as follows:

$\left\{\begin{array}{c}\stackrel{.}{e}\left(t\right)=(W_1+N{B}_{1}E)e\left(t\right)+N{B}_2{d}_2\left(t\right)+H_1\stackrel{.}{F}\left(t\right)+H_3\mathrm{\δ}\left(t\right)\\ z_{\∞}\left(t\right)=\mathrm{Ce}\left(t\right)\end{array}\right.$

wherein $e\left(t\right)=\left[\begin{array}{c}{e}_w\left(t\right)\\ e_F\left(t\right)\end{array}\right],$ $W_1=\left[\begin{array}{cc}W& 0\\ 0& 0\end{array}\right],$ $N=\left[\begin{array}{c}L\\ K\end{array}\right],$ E＝[VI]， $H_1=\left[\begin{array}{c}0\\ I\end{array}\right],$ $H_3=\left[\begin{array}{c}{B}_3\\ 0\end{array}\right].$

z_∞(t) is H_∞Performance reference output, C is H_∞And (5) outputting a matrix with adjustable performance.

Solving a modelable interference observer gain matrix and a fault diagnosis observer gain matrix of the multi-source interference system by using a convex optimization algorithm; given an initial value e_w(0) And e_F(0) Adjustable transfusionOut of matrix C, interference suppression degree gamma₁、γ_２And gamma_３Solving the following convex optimization problem:

min(e^T(0)Pe(0))

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}\mathrm{sym}(P{W}_1+R{B}_{1}E)& P{H}_3& P{H}_1& R{B}_2& C^T\\ *& {{-\mathrm{\&gamma;}}_1}^{2}I& 0& 0& 0\\ *& *& -{{\mathrm{\&gamma;}}_2}^{2}I& 0& 0\\ *& *& *& -{{\mathrm{\&gamma;}}_3}^{2}I& 0\\ *& *& *& *& -I\end{array}\right]<0$

wherein symbols denote symmetric blocks, sym (PW) of the corresponding part of the symmetric matrix₁+RB₁E) The expression is as follows: sym (PW)₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T。

Solve the above equation to obtain P, R, the observer gain matrix $\left[\begin{array}{c}L\\ K\end{array}\right]=P^{-1}R.$

Fourthly, designing a sliding mode controller, and respectively compensating the fault and the disturbance by using the estimated values of the fault and the disturbance, wherein the design steps of the sliding mode controller are as follows:

1) design sliding mode surface s (t)

The general design method of the slip form face is as follows:

$s\left(t\right)=\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}{k}_i{x}_i\left(t\right)+x_n\left(t\right),$ wherein k is_i＞0，i＝1，2，…，n-1。

2) Design sliding mode control law

Adopting function switching control law, including equivalent input and switching input, the equivalent input is formed fromAnd (6) obtaining. The control law is designed as follows:

u(t)＝u_eq(t)+u_vs(t)

wherein u is_eq(t) is the equivalent control quantity of the system, u_vsAnd (t) is a switching control amount.

Order toIs provided withThe kinetic model of the substitution system can be obtained

$-a_0{x}_n\left(t\right)-\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;-a_{n-1}{x}_1\left(t\right)+b_1(u\left(t\right)+d_1\left(t\right)+F\left(t\right))+b_2{d}_2\left(t\right)=-\underset{i=1}{\overset{n-1}{\mathrm{\&Sigma;}}}{k}_i{\stackrel{.}{x}}_i\left(t\right)$

U (t) obtained from the above equation is an equivalent control amount, and further:

$u_{\mathrm{eq}}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\mathrm{\&Sigma;}}}{k}_{i-1}{x}_i\left(t\right))-d_1\left(t\right)-F\left(t\right);$

using estimates of modelable disturbances and faultsRespectively replacing the actual value d₁(t), F (t) is obtained $u_{\mathrm{eq}}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\mathrm{\&Sigma;}}}{k}_{i-1}{x}_i\left(t\right))-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right);$

The switch control quantity is designed as u_vs(t)＝-T_psgn (s (t)). Wherein, T_pObtaining the sliding mode gain through the fifth step; sgn (s (t)) is a switching function and is expressed in the following form:

$\mathrm{sgn}\left(s\left(t\right)\right)=\left\{\begin{array}{cc}1& s\left(t\right)>0\\ 0& s\left(t\right)=0\\ -1& s\left(t\right)<0\end{array}\right.$

the control input expression is:

$u\left(t\right)=u_{\mathrm{eq}}\left(t\right)+u_{\mathrm{vs}}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\mathrm{\&Sigma;}}}{k}_{i-1}{x}_i\left(t\right))-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right)-T_p\mathrm{sgn}\left(s\left(t\right)\right)$

fifthly, solving the sliding mode gain to ensure the system stability

The Lyapunov function is designed as

From the definition of s (t) and the kinetic model of the system, it can be obtained

$\stackrel{.}{s}\left(t\right)=b_1(u\left(t\right)+d_1\left(t\right)+F\left(t\right))+b_2{d}_2\left(t\right)-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{n-i}{x}_i\left(t\right)+\underset{i=2}{\overset{n}{\mathrm{\&Sigma;}}}{k}_{i-1}{x}_i\left(t\right)$

Substituting the control input expression obtained in the fourth step into the formula

$\stackrel{.}{G}\left(t\right)=s^T\left(t\right)\stackrel{.}{s}\left(t\right)=s^T\left(t\right)(b_1{e}_F\left(t\right)+b_{1}V{e}_w\left(t\right)+b_2{d}_2\left(t\right)-T_p\mathrm{sgn}\left(s\left(t\right)\right))$

According to Lyapunov's theorem, whenIt is true that the system is proven to be able to reach the slip mode plane, and the slip mode plane is asymptotically stable, note α ═ b₁e_F(t)+b₁Ve_w(t)+b₂d₂(T) |, it is clear that T needs to be satisfied_pNot less than α, there areThe system can reach the sliding mode surface and reach the asymptotic consistent stable state.

Taking into account the saturated input problem of the system, T_p＝max(α，u_om). Wherein u is_omIs the value of the saturated input to the system, $\max (\mathrm{\&alpha;},u_{\mathrm{om}})=\left\{\begin{array}{cc}\mathrm{\&alpha;}& \mathrm{\&alpha;}>u_{\mathrm{om}}\\ u_{\mathrm{om}}& \mathrm{\&alpha;}\&le;u_{\mathrm{om}}\end{array}.\right.$

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

the sliding mode control method with the anti-interference fault-tolerant performance adopts the combination of the sliding mode controller, the fault diagnosis observer and the interference observer, and is suitable for the anti-interference fault-tolerant control of an input saturation system. The fault diagnosis observer and the disturbance observer ensure the fault-tolerant anti-interference performance of the system, the fault and the disturbance are estimated and counteracted, the sliding mode controller has robustness to the disturbance, and the design of the sliding mode gain takes the saturated input value of the system into consideration. The control method solves the fault-tolerant anti-interference problem of the input saturation system. Meanwhile, two observers are designed to counteract faults and interference in the system, so that the buffeting phenomenon of sliding mode control is weakened, and the precision and reliability of the sliding mode control method in the attitude control system are improved.

Drawings

Fig. 1 is a design flow chart of a sliding mode control method with anti-interference fault-tolerant performance according to the present invention.

Detailed Description

As shown in fig. 1, the implementation steps of the present invention are as follows (taking a three-axis stable satellite attitude control system as an example to illustrate the implementation of the method):

1. considering faults and multi-source interference in satellite attitude control system, and building system dynamics model

The Euler angle between the body coordinate system and the orbit coordinate system of the three-axis stable satellite is very small, and the satellite attitude dynamics and the kinematics model are linearized to obtain:

$\left\{\begin{array}{c}{J}_1\stackrel{..}{\mathrm{\&phi;}}-{\mathrm{\&omega;}}_0(J_1-J_2+J_3)\stackrel{.}{\mathrm{\&psi;}}+4{{\mathrm{\&omega;}}_0}^2(J_2-J_3)\mathrm{\&phi;}=u_1+T_{d1}\\ J_2\stackrel{..}{\mathrm{\&theta;}}+3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)\mathrm{\&theta;}=u_2+F\left(t\right)+T_{d2}\\ J_3\stackrel{..}{\mathrm{\&psi;}}+{\mathrm{\&omega;}}_0(J_1-J_2+J_3)\stackrel{.}{\mathrm{\&phi;}}+{{\mathrm{\&omega;}}_0}^2(J_2-J_1)\mathrm{\&psi;}=u_3+T_{d3}\end{array}\right.$

the three equations of the above formula are attitude dynamics equations of the satellite roll shaft, the pitch shaft and the yaw shaft in three axial directions in sequence. J. the design is a square₁，J₂，J₃The three-axis rotational inertia is phi, theta and psi respectively are three-axis Euler angles between a satellite body coordinate system and an orbit coordinate system;respectively, the three-axis euler angular rates;respectively, three-axis euler angular acceleration; u. of₁，u₂，u₃Three-axis control moment respectively; omega₀Is the satellite orbit angular velocity; f (T) is a time-varying fault, T_d1，T_d2，T_d3Interference moments (including interference moments brought by a sensor and an actuating mechanism) which are three axes respectively;

the following steps take a satellite pitching channel dynamic model as an example to design a sliding mode controller with anti-interference fault-tolerant performance, and the design method of a roll shaft is the same as that of a yaw shaft.

The expected attitude angle, angular velocity and angular acceleration of the three-axis stabilized satellite are respectively recorded as theta_c(t)、Andand are all zero. The available error system equation is as follows:

$J_2{\stackrel{..}{e}}_{\mathrm{\&theta;}}+3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)e_{\mathrm{\&theta;}}=u_2+F\left(t\right)+T_{d2}$

wherein e_θ(t)＝θ(t)-θ_cWhere θ (t) is an error angle,is the error angular acceleration.

Writing the pitch channel attitude error model into a state space form as follows:

$\stackrel{.}{X}\left(t\right)=\mathrm{AX}\left(t\right)+B_1(u\left(t\right)+d_1\left(t\right)+F\left(t\right))+B_2{d}_2\left(t\right)$

wherein the state variables of the multisource interference systemu (t) is control input, F (t) is time-varying fault, d₁(t) modelable interference, d₂(t) unmodeled interference, d₁(t) and d₂(T) composition T_d2。A、B₁And B₂As follows:

$A=\left[\begin{array}{cc}0& 1\\ -3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)/J_2& 0\end{array}\right],$ $B_1=\left[\begin{array}{c}0\\ J_2^{-1}\end{array}\right],$ $B_2=\left[\begin{array}{c}0\\ J_2^{-1}\end{array}\right].$

external model description interference d₁(t) from the following external interference model ∑₁Represents:

${\mathrm{\&Sigma;}}_1:\left\{\begin{array}{c}{d}_1\left(t\right)=\mathrm{Vw}\left(t\right)\\ \stackrel{.}{w}\left(t\right)=\mathrm{Ww}\left(t\right)+B_3\mathrm{\&delta;}\left(t\right)\end{array}\right.$

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 (t) is an energy-bounded, unmodeled random (i.e., L)₂Norm ofBounded) interference, B₃A gain array that is unmoldable for interference.

2. Designing a fault diagnosis observer and a disturbance observer to respectively estimate faults and modelable disturbance

Designing a fault diagnosis observer aiming at the time-varying fault F (t) in the system as follows:

$\left\{\begin{array}{c}\hat{F}\left(t\right)=\mathrm{\&xi;}\left(t\right)-\mathrm{KX}\left(t\right))\\ \stackrel{.}{\mathrm{\&xi;}}\left(t\right)=K{B}_1(\mathrm{\&xi;}\left(t\right)-\mathrm{KX}\left(t\right))+K[\mathrm{AX}\left(t\right)+B_{1}u\left(t\right)+B_1\widehat{d_1}\left(t\right))]\end{array}\right.$

for modelable interference d in a system₁(t) designing a disturbance observer as follows:

$\left\{\begin{array}{c}\widehat{d_1}\left(t\right)=V\hat{w}\left(t\right)\\ \hat{w}\left(t\right)=r\left(t\right)-\mathrm{LX}\left(t\right)\\ \stackrel{.}{r}\left(t\right)=(W+L{B}_{1}V)(r\left(t\right)-\mathrm{LX}\left(t\right))+L[\mathrm{AX}\left(t\right)+B_{1}u\left(t\right)+B_1\hat{F}\left(t\right)]\end{array}\right.$

wherein,is an estimate of the fault that is,in order to be an interference estimate,and (d) an estimated value of w (t), ξ (t) and r (t) are auxiliary variables in the fault diagnosis observer and the disturbance observer respectively, and K and L are a gain matrix of the fault diagnosis observer to be determined and a gain matrix of the disturbance observer respectively, and are obtained by the subsequent step 3.

Defining fault estimation errors $e_F\left(t\right)=F\left(t\right)-\hat{F}\left(t\right),$ Interference estimation error $e_w\left(t\right)=w\left(t\right)-\hat{w}\left(t\right);$

According to the expression of the fault diagnosis observer, the fault estimation error equation can be obtained as follows:

${\stackrel{.}{e}}_F\left(t\right)=K{B}_1{e}_F\left(t\right)+K{B}_{1}V{e}_w\left(t\right)+K{B}_2{d}_2\left(t\right)+\stackrel{.}{F}\left(t\right)$

according to the expression of the disturbance observer, the disturbance estimation error equation can be obtained as follows:

${\stackrel{.}{e}}_w\left(t\right)=L{B}_{1}V{e}_w\left(t\right)+L{B}_1{e}_F\left(t\right)+L{B}_2{d}_2\left(t\right)+B_3\mathrm{\&delta;}\left(t\right)$

3. fault diagnosis observer gain matrix and interference observer gain matrix solving

The estimation error equation for modelable interference and the estimation error equation for fault in the second step of the cascade are as follows:

$\left\{\begin{array}{c}\stackrel{.}{e}\left(t\right)=(W_1+N{B}_{1}E)e\left(t\right)+N{B}_2{d}_2\left(t\right)+H_1\stackrel{.}{F}\left(t\right)+H_3\mathrm{\&delta;}\left(t\right)\\ z_{\&infin;}\left(t\right)=\mathrm{Ce}\left(t\right)\end{array}\right.$

wherein $e\left(t\right)=\left[\begin{array}{c}{e}_w\left(t\right)\\ e_F\left(t\right)\end{array}\right],$ $W_1=\left[\begin{array}{cc}W& 0\\ 0& 0\end{array}\right],$ $N=\left[\begin{array}{c}L\\ K\end{array}\right],$ E＝[VI]， $H_1=\left[\begin{array}{c}0\\ I\end{array}\right],$ $H_3=\left[\begin{array}{c}{B}_3\\ 0\end{array}\right].$ I is a unit array, z_∞(t) is H_∞Performance reference output, C is H_∞A performance output matrix.

Solving a fault-tolerant anti-interference controller gain array of the multi-source interference system by using a convex optimization algorithm; given an initial value e_w(0) And e_F(0) Output matrix C, interference suppression degree gamma₁、γ₂And gamma₃Solving the following convex optimization problem:

min(e^T(0)Pe(0))

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}\mathrm{sym}(P{W}_1+R{B}_{1}E)& P{H}_3& P{H}_1& R{B}_2& C^T\\ *& {{-\mathrm{\&gamma;}}_1}^{2}I& 0& 0& 0\\ *& *& -{{\mathrm{\&gamma;}}_2}^{2}I& 0& 0\\ *& *& *& -{{\mathrm{\&gamma;}}_3}^{2}I& 0\\ *& *& *& *& -I\end{array}\right]<0$

in the above formula, symbols denote symmetric blocks, sym (PW) of the corresponding portion in the symmetric matrix₁+RB₁E) The expression is as follows: sym (PW)₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T。

Solve the above equation to obtain P, R, the observer gain matrix $\left[\begin{array}{c}L\\ K\end{array}\right]=P^{-1}R.$

4. Designing sliding mode controller to compensate for fault and disturbance separately using estimated values of fault and disturbance

1) The switching function is designed as follows:

$s\left(t\right)=e_{\mathrm{\&theta;}}\left(t\right)+{\stackrel{.}{e}}_{\mathrm{\&theta;}}\left(t\right)$

wherein e is_θ(t)＝θ(t)-θ_c(t), three-axis stabilized satellite desired attitude angle θ_c(t) is 0, therefore, there is

$s\left(t\right)=\mathrm{\&theta;}\left(t\right)+\stackrel{.}{\mathrm{\&theta;}}\left(t\right)$

2) The sliding mode control law is designed as follows:

u(t)＝u_eq(t)+u_vs(t)

wherein u is_eq(t) is an equivalent controlled variable, u_vsAnd (t) is a switching control amount.

Order toIs provided withSubstituting the kinetic equation of the pitch axis to obtain

$-J_2{k}_1\stackrel{.}{\mathrm{\&theta;}}\left(t\right)=-3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)\mathrm{\&theta;}\left(t\right)+u_{\mathrm{eq}}\left(t\right)+d_1\left(t\right)+F\left(t\right).$

Using estimates of modelable disturbances and faultsRespectively replacing the actual value d₁(t), F (t) is obtained $u_{\mathrm{eq}}\left(t\right)=3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)\mathrm{\&theta;}\left(t\right)-J_2{k}_1\stackrel{.}{\mathrm{\&theta;}}\left(t\right)-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right).$

The switch control quantity is designed as u_vs(t)＝-T_psgn (s (t)). Wherein T is_pFor the sliding mode gain, which is determined in the following step 5, sgn (s (t)) is a switching function and is expressed in the following form:

$\mathrm{sgn}\left(s\left(t\right)\right)=\left\{\begin{array}{cc}1& s\left(t\right)>0\\ 0& s\left(t\right)=0\\ -1& s\left(t\right)<0\end{array}\right.$

the system control inputs are as follows:

$u\left(t\right)=u_{\mathrm{eq}}\left(t\right)+u_{\mathrm{vs}}\left(t\right)=3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)\mathrm{\&theta;}\left(t\right)-J_2{k}_1\stackrel{.}{\mathrm{\&theta;}}\left(t\right)-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right)-T_p\mathrm{sgn}\left(s\left(t\right)\right)$

5. solving the sliding mode gain and ensuring the stability of the system

The Lyapunov function is designed as $G\left(t\right)=\frac{1}{2}{s}^T\left(t\right)J_{2}s\left(t\right)\&GreaterEqual;0.$

By definition $\stackrel{.}{s}\left(t\right)=k_1\stackrel{.}{\mathrm{\&theta;}}\left(t\right)+\stackrel{..}{\mathrm{\&theta;}}\left(t\right)=k_1\stackrel{.}{\mathrm{\&theta;}}\left(t\right)+(-3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)\mathrm{\&theta;}\left(t\right)+u\left(t\right)+d_1\left(t\right)+F\left(t\right))/J_2$

Substituting the control input expression into the above formula to obtain

$\stackrel{.}{G}\left(t\right)=s^T\left(t\right)J_2\stackrel{.}{s}\left(t\right)=s^T\left(t\right)(b_1{e}_F\left(t\right)+b_{1}V{e}_w\left(t\right)+b_2{d}_2\left(t\right)-T_p\mathrm{sgn}\left(s\left(t\right)\right))$

According to Lyapunov's theorem, whenThe establishment proves that the system can reach the sliding mode surface, and the sliding mode plane is asymptotically stable;

remember α | | b₁e_F(t)+b₁Ve_w(t)+b₂d₂(t) |. Obviously, T needs to be satisfied_pNot less than α, there areThe system can reach the sliding mode surface and reach the asymptotic consistent stable state.

Taking into account the saturated input problem of the system, T_p＝max(α，u_om). Wherein u is_omIt is known that the maximum input torque provided to the flywheel, $\max (\mathrm{\&alpha;},u_{\mathrm{om}})=\left\{\begin{array}{cc}\mathrm{\&alpha;}& \mathrm{\&alpha;}>u_{\mathrm{om}}\\ u_{\mathrm{om}}& \mathrm{\&alpha;}\&le;u_{\mathrm{om}}\end{array}\right..$

in order to avoid the buffeting phenomenon of the output of the sliding mode controller, a saturation function sat (s (t)) is adopted to replace a switch function sgn (s (t)). The expression sat (s (t)) is as follows:

$\mathrm{sat}\left(s\left(t\right)\right)=\left\{\begin{array}{cc}\mathrm{sgn}\left(s\left(t\right)\right)& \left|s\left(t\right)\right|>\mathrm{\&sigma;}\\ s\left(t\right)/\left|\mathrm{\&sigma;}\right|& \left|s\left(t\right)\right|\&le;\mathrm{\&sigma;}\end{array}\right.$

wherein, sigma is a flutter eliminating factor, which aims to effectively eliminate buffeting and ensure the rapid convergence of the system, and the value is in the range of (0.02, 0.08).

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

Claims (1)

1. A sliding mode control method with anti-interference fault-tolerant performance is characterized by comprising the following steps: firstly, establishing a dynamic model of a system; secondly, designing a fault diagnosis observer and a disturbance observer to estimate faults and modelable disturbance in a system; thirdly, solving a gain matrix of the interference observer and the fault diagnosis observer; then designing a sliding mode controller, substituting the interference and fault estimation values into the sliding mode controller to compensate equivalent interference and faults; finally, solving the sliding mode gain on the premise of system input saturation to ensure the stability of the system; the method comprises the following specific steps:

firstly, establishing a system dynamics model

A system dynamics model containing interference and faults is built, and the following steps are included:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1\left(t\right)=x_2\left(t\right)\\ {\stackrel{\&CenterDot;}{x}}_2\left(t\right)=x_3\left(t\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ {\stackrel{\&CenterDot;}{x}}_n\left(t\right)=-a_0{x}_n\left(t\right)-......-a_{n-1}{x}_1\left(t\right)+b_1\left(u\right(t)+d_1\left(t\right)+F\left(t\right))+b_2{d}_2\left(t\right)\end{array}\right.$

wherein x is₁(t),x₂(t),…,x_n(t) is the system state, n ≧ 2 is a positive integer, u (t) is the control input, F (t) is the rate-of-change bounded time-varying fault, d₁(t) modelable interference, d₂(t) unmoldable random interference, a₀、a₁、…a_n-1And b₁、b₂Are all system internal parameters; d₁(t) from the following external interference model ∑₁Represents:

${\mathrm{\&Sigma;}}_1:\left\{\begin{array}{c}{d}_1\left(t\right)=Vw\left(t\right)\\ \stackrel{\&CenterDot;}{w}\left(t\right)=Ww\left(t\right)+B_3\mathrm{\&delta;}\left(t\right)\end{array}\right.$

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, B₃A gain array that is unmoldable random interference, (t) unmoldable random interference that is energy bounded;

selecting a system state variable X (t) ═ x₁(t)x₂(t)……x_n(t)]^TWritten as a state space expression is as follows:

$\stackrel{\&CenterDot;}{X}\left(t\right)=AX\left(t\right)+B_1\left(u\right(t)+d_1(t)+F(t\left)\right)+B_2{d}_2\left(t\right)$

wherein X (t) is system state variable, A is system array, B₁As an input matrix, B₂A gain array for unmoldable random interference;

designing a fault diagnosis observer and a disturbance observer to respectively estimate faults and modelable disturbance

Designing a fault diagnosis observer aiming at the time-varying fault F (t) in the system as follows:

$\left\{\begin{array}{c}\hat{F}\left(t\right)=\mathrm{\&xi;}\left(t\right)-KX\left(t\right)\\ \stackrel{\&CenterDot;}{\mathrm{\&xi;}}\left(t\right)={\mathrm{KB}}_1\left(\mathrm{\&xi;}\right(t)-KX(t\left)\right)+K\&lsqb;AX\left(t\right)+B_{1}u\left(t\right)+B_1{\hat{d}}_1\left(t\right)\&rsqb;\end{array}\right.$

for modelable interference d in a system₁(t) designing a disturbance observer as follows:

$\left\{\begin{array}{c}{\hat{d}}_1\left(t\right)=V\hat{w}\left(t\right)\\ \hat{w}\left(t\right)=r\left(t\right)-LX\left(t\right)\\ \stackrel{\&CenterDot;}{r}\left(t\right)=(W+L{B}_{1}V)\left(r\right(t)-LX(t\left)\right)+L\&lsqb;AX\left(t\right)+B_{1}u\left(t\right)+B_1\hat{F}\left(t\right)\&rsqb;\end{array}\right.$

wherein,is an estimate of the fault that is,in order to be an interference estimate,the estimated value of w (t), ξ (t) and r (t) are respectively auxiliary variables in the fault diagnosis observer and the disturbance observer, K and L are respectively a gain matrix of the fault diagnosis observer to be determined and a gain matrix of the disturbance observer, and the estimated value is obtained by the subsequent third step;

defining a fault estimation error asInterference observation error is $e_w\left(t\right)=w\left(t\right)-\hat{w}\left(t\right);$

The fault estimation error equation can be obtained according to the expression of the fault diagnosis observer as follows:

${\stackrel{\&CenterDot;}{e}}_F\left(t\right)={\mathrm{KB}}_1{e}_F\left(t\right)+{\mathrm{KB}}_1{\mathrm{Ve}}_w\left(t\right)+{\mathrm{KB}}_2{d}_2\left(t\right)+\stackrel{\&CenterDot;}{F}\left(t\right)$

the disturbance estimation error equation can be derived from the expression of the disturbance observer as follows:

${\stackrel{\&CenterDot;}{e}}_w\left(t\right)={\mathrm{LB}}_1{\mathrm{Ve}}_w\left(t\right)+{\mathrm{LB}}_1{e}_F\left(t\right)+{\mathrm{LB}}_2{d}_2\left(t\right)+B_3\mathrm{\&delta;}\left(t\right)$

thirdly, solving the gain matrix of the fault diagnosis observer and the gain matrix of the interference observer

The interference estimation error equation and the fault estimation error equation in the second step of the cascade are as follows:

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{e}\left(t\right)=(W_1+{\mathrm{NB}}_{1}E)e\left(t\right)+{\mathrm{NB}}_2{d}_2\left(t\right)+H_1\stackrel{\&CenterDot;}{F}\left(t\right)+H_3\mathrm{\&delta;}\left(t\right)\\ z_{\mathrm{\&infin;}}\left(t\right)=Ce\left(t\right)\end{array}\right.$

wherein $e\left(t\right)=\left[\begin{array}{c}{e}_w\left(t\right)\\ e_F\left(t\right)\end{array}\right],W_1=\left[\begin{array}{cc}W& 0\\ 0& 0\end{array}\right],N=\left[\begin{array}{c}L\\ K\end{array}\right],E=\left[\begin{array}{cc}V& I\end{array}\right],H_1=\left[\begin{array}{c}0\\ I\end{array}\right],H_3=\left[\begin{array}{c}{B}_3\\ 0\end{array}\right];$ I is a unit array, z_∞(t) is H_∞Performance reference output, C is H_∞A performance adjustable output matrix;

solving a modelable interference observer gain matrix and a fault diagnosis observer gain matrix of the multi-source interference system by using a convex optimization algorithm; given an initial value e_w(0) And e_F(0) Adjustable output matrix C, interference suppression degree gamma₁、γ₂And gamma₃Solving the following convex optimization problem:

min(e^T(0)Pe(0))

$\mathrm{\&Phi;}=\left[\begin{array}{ccccc}sym({\mathrm{PW}}_1+{\mathrm{RB}}_{1}E)& {\mathrm{PH}}_3& {\mathrm{PH}}_1& {\mathrm{RB}}_2& C^T\\ *& -{{\mathrm{\&gamma;}}_1}^2& 0& 0& 0\\ *& *& -{\mathrm{\&gamma;}}_2^2& 0& 0\\ *& *& *& -{\mathrm{\&gamma;}}_3^2& 0\\ *& *& *& *& -I\end{array}\right]<0$

wherein symbols denote symmetric blocks, sym (PW) of the corresponding part of the symmetric matrix₁+RB₁E) The expression is as follows:

sym(PW₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T；

p, R, observer gain matrix, is solved for the above equation $\left[\begin{array}{c}L\\ K\end{array}\right]=P^{-1}R;$

Fourthly, designing a sliding mode controller, and respectively compensating the fault and the disturbance by using the estimated values of the fault and the disturbance, wherein the design steps of the sliding mode controller are as follows:

1) design sliding mode surface s (t)

The design method of the sliding mode surface comprises the following steps:

$s\left(t\right)=\underset{i=1}{\overset{n-1}{\&Sigma;}}{k}_i{x}_i\left(t\right)+x_n\left(t\right),$ wherein k is_i＞0,i＝1,2,…,n-1；

2) Design sliding mode control law

Adopting function switching control law, including equivalent input and switching input, the equivalent input is formed fromObtaining; the control law is designed as follows:

u(t)＝u_eq(t)+u_vs(t)

wherein u is_eq(t) is the equivalent control quantity of the system, u_vs(t) is a switching control quantity;

order toIs provided withThe kinetic model of the substitution system can be obtained

$-a_0{x}_n\left(t\right)-......-a_{n-1}{x}_1\left(t\right)+b_1\left(u\right(t)+d_1(t)+F(t\left)\right)+b_2{d}_2\left(t\right)=-\underset{i=1}{\overset{n-1}{\&Sigma;}}{k}_i{\stackrel{\&CenterDot;}{x}}_i\left(t\right)$

U (t) obtained from the above equation is an equivalent control amount, and further:

$u_{eq}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\&Sigma;}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\&Sigma;}}{k}_{i-1}{x}_i\left(t\right))-d_1\left(t\right)-F\left(t\right);$

using estimates of modelable disturbances and faultsRespectively replacing the actual value d₁(t), F (t), have $u_{eq}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\&Sigma;}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\&Sigma;}}{k}_{i-1}{x}_i\left(t\right))-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right);$

The switch control quantity is designed as u_vs(t)＝-T_psgn (s (t)); wherein, T_pObtaining the sliding mode gain through the fifth step; sgn (s (t)) is a switching function and is expressed in the following form:

$sgn\left(s\right(t\left)\right)=\left\{\begin{array}{cc}1& s\left(t\right)>0\\ 0& s\left(t\right)=0\\ -1& s\left(t\right)<0\end{array}\right.$

the control input expression is:

$u\left(t\right)=u_{eq}\left(t\right)+u_{vs}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\&Sigma;}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\&Sigma;}}{k}_{i-1}{x}_i\left(t\right))-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right)-T_p\mathrm{sgn}\left(s\right(t\left)\right)$

fifthly, solving the sliding mode gain to ensure the system stability

The Lyapunov function is designed as

From the definition of s (t) and the kinetic model of the system, it can be obtained

$\stackrel{\&CenterDot;}{s}\left(t\right)=b_1\left(u\right(t)+d_1(t)+F(t\left)\right)+b_2{d}_2\left(t\right)-\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{n-i}{x}_i\left(t\right)+\underset{i=2}{\overset{n}{\mathrm{\&Sigma;}}}{k}_{i-1}{x}_i\left(t\right)$

Substituting the control input expression obtained in the fourth step into the formula

$\stackrel{\&CenterDot;}{G}\left(t\right)=s^T\left(t\right)\stackrel{\&CenterDot;}{s}\left(t\right)=s^T\left(t\right)\left(b_1{e}_F\right(t)+b_1{\mathrm{Ve}}_w(t)+b_2{d}_2(t)-T_p\mathrm{sgn}(s\left(t\right)\left)\right)$

According to Lyapunov's theorem, whenThe establishment proves that the system can reach the sliding mode surface, and the sliding mode plane is asymptotically stable;

remember α | | b₁e_F(t)+b₁Ve_w(t)+b₂d₂(T) |, it is obvious that only T needs to be satisfied_pNot less than α, there areThe system can reach the sliding mode surface and reach a stable state of asymptotic consistency;

considering the saturated input problem of the system, the switching value T_p＝max(α,u_om) (ii) a Wherein u is_omIs the value of the saturated input to the system, $max(\mathrm{\&alpha;},u_{om})=\left\{\begin{array}{cc}\mathrm{\&alpha;}& \mathrm{\&alpha;}>u_{om}\\ u_{om}& \mathrm{\&alpha;}\&le;u_{om}\end{array}\right..$