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

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

Technical field

The present invention relates to a kind of sliding-mode control with anti-interference fault freedom, the method can be used for defeatedEnter the anti-interference fault-tolerant control of constrained system, as be subject to flywheel maximum (top) speed restriction input torque satellite, flyThe attitude RACS in the Aero-Space such as machine and survey of deep space field.

Background technology

Complicated along with spacecraft task, also more and more higher to the requirement of attitude control accuracy, spacecraftHigh-precision attitude control become domestic and international study hotspot. The space environment complexity of spacecraft operation, is subject toExtraneous and inner multi-source disturbs and modeling is not dynamic, and attitude control system fault rate is higher.The fault of sensor, flywheel etc. can cause tasks interrupt even to lose efficacy, for improving the reliability of system, necessaryAdopt accurately fault diagnosis and fault-tolerant control method. Meanwhile, not modeling dynamically, unknown parameter, randomThe many factors such as interference and other disturbance variables of equal value cause attitude of flight vehicle modeling of control system inaccuracy,Cause the control accuracy even unstability that declines, the anti-interference attitude control method of spacecraft is extremely important. In addition,Flywheel is subject to the restriction of maximum (top) speed, the torque input constraint of generation, and there is the saturated problem of input in system.Input the saturated control performance that can affect system, easily cause system unstability, must be in the design of controllerIn journey, take in.

For the problems referred to above, Chinese scholars has proposed a lot of effective methods. When existing and only deposit in systemAccident barrier in the situation that in the time of derivative bounded, method for diagnosing faults be divided into method based on dynamic mathematical models,Method based on signal processing and the method based on knowledge, wherein the method based on observer, based on nerve netThe method of network and the research of wavelet transformation are very extensive. In system, only exist in situation about can modeling disturbing,Control based on interference observer can and be offset Interference Estimation, and advantage is simple in structure, for systemDifferent performance requires can be in conjunction with different control methods. Above method all can not be applied to and have fault, dryDisturb and input in the system control of saturated restriction.

In recent years, sliding formwork control is subject to more and more higher attention, the method because of its good characteristic havingInsensitive, simple in structure to parameter variation and disturbance, be applicable to the control of satellite attitude control system. Due toFault in attitude control system and interference effect, very easily there is chattering phenomenon in sliding-mode control, has before thisScholar designs interference observer and offsets the impact of disturbing, and improves the buffeting problem of sliding formwork control, realizes highly reliableProperty and high-precision attitude control, but do not consider fault.

Summary of the invention

Technology of the present invention is dealt with problems and is: for the saturated multi-source EVAC of input, propose one and haveThe sliding-mode control of Fault Compensation and Interference Cancellation and rejection. By design error failure diagnostic observations device withInterference observer, estimates and offsets, design sliding mode controller is to parameter to the fault in system and interferenceUncertain and disturbance has robustness.

Technical solution of the present invention is: a kind of sliding-mode control with anti-interference fault freedom, itsBe characterised in that and comprise the following steps:

First, the fault in taking into account system and multi-source disturb, and set up the kinetic model of system; Secondly,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, willIn interference and Fault Estimation value substitution controller, compensation is of equal value disturbs and fault; Finally, analyzer-controller is steadyQualitative, input and under saturated prerequisite, solve sliding formwork gain in system; Concrete steps are as follows:

The first step, the fault in taking into account system and multi-source disturb, and set up the kinetic model of system

Build the system dynamics model that comprises fault and interference, 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, x₁(t)，x₂(t)，…，x_n(t) be system mode, n >=2 are positive integer, and u (t) is control inputs, F (t)For the time accident of rate of change bounded hinders, d₁(t) be modeling to disturb, d₂(t) being can not modeling random disturbances. a₀、a₁、…a_n-1With b₁、b₂Be internal system parameter. d₁(t) can be by following interference model ∑₁Represent:

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

Wherein, w (t) be can modeling interference model state variable, V be can modeling interference model output matrix,W represents system battle array that can modeling interference model, B₃For gain battle array that can not modeling random disturbances, δ (t) isEnergy bounded can not modeling random disturbances.

Choose state variable X (t)=[x₁(t)x₂(t)......x_n(t)]^T, write as state-space expression 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 variables, and A is system battle array, B₁For input matrix, B₂For can not modelingThe gain battle array of random disturbances.

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

Second step, design error failure diagnostic observations device and interference observer are distinguished suspected fault and can modeling be disturbedHindering F (t) design error failure diagnostic observations device for the time accident in system is:

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

For the d that can modeling disturbs in system₁(t) design interference observer is:

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

Wherein,For the estimated value to fault,For Interference Estimation value,For the estimated value of w (t),ξ (t) and r (t) are respectively the auxiliary variable in fault diagnosis observer and interference observer, and K and L are respectivelyFault diagnosis observer gain matrix undetermined and interference observer gain matrix, tried to achieve by subsequent step 3.

Failure definition evaluated error $e_F\left(t\right)=F\left(t\right)-\hat{F}\left(t\right),$ Disturbance-observer error $e_w\left(t\right)=w\left(t\right)-\hat{w}\left(t\right);$

Can obtain Fault Estimation error equation according to the expression formula of fault diagnosis observer is:

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

Can obtain Interference Estimation error equation according to the expression formula of interference observer is:

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

The 3rd step, fault diagnosis observer gain matrix and interference observer gain matrix solve

The evaluated error equation can modeling disturbing in connection row second step and the evaluated error equation of fault asUnder:

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

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) be H_∞Performance is with reference to output, and C is H_∞Performance is adjustable output matrix.

Utilize convex optimized algorithm solve multi-source EVAC can modeling interference observer gain matrix and fault examineDisconnected observer gain matrix; Given initial value e_wAnd e (0)_F(0), adjustable output matrix C, disturbs inhibition degreeγ₁、γ_２And γ_３, solve following protruding 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, symbol * represents the symmetry blocks of appropriate section in symmetrical matrix, sym (PW₁+RB₁E) expression formula asUnder: sym (PW₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T。

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

The 4th step, design sliding mode controller, the estimated value of operational failure and interference is compensate for failed and interference respectivelyThe design procedure of sliding mode controller is as follows:

1) design sliding-mode surface s (t)

The common method for designing of sliding-mode surface 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_i＞0，i＝1，2，…，n-1。

2) design sliding formwork control law

Adopt function switching law, comprise input of equal value and switch input two parts, input of equal value byTry to achieve. Design of control law is as follows:

u(t)＝u_eq(t)+u_vs(t)

Wherein u_eq(t) be the controlled quentity controlled variable of equal value of system, u_vs(t) be switch controlled quentity controlled variable.

OrderHaveThe kinetic model of substitution system can obtain

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

The u (t) being obtained by above formula is controlled quentity controlled variable of equal value, and then can obtain:

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

Use can modeling interference and the estimated value of faultReplace respectively actual value d₁(t)、F(t)，Try to achieve $u_{\mathrm{eq}}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\mathrm{\&Sigma;}}}{k}_{i-1}{x}_i\left(t\right))-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right);$

Switch controlled quentity controlled variable is designed to u_vs(t)＝-T_pSgn (s (t)). Wherein, T_pFor sliding formwork gain, by the 5th stepTry to achieve; Sgn (s (t)) is switch function, represents by following form:

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

Control inputs expression formula is:

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

The 5th step, solves sliding formwork gain, ensures system stability

Liapunov function is designed to

By the definition of s (t) and the kinetic model of system, can obtain

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

The control inputs that the 4th step is tried to achieve is expressed formula substitution above formula, has

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

According to Lyapunov theorem, whenSet up, proof system can reach sliding-mode surface, and slidingDynamic model state plane is asymptotically stable. Note α=|| b₁e_F(t)+b₁Ve_w(t)+b₂d₂(t) ||, obviously, need to meetT_p>=α, hasSystem can reach sliding-mode surface, and reaches asymptotic consistent stable state.

Consider the saturated input problem of system, T_p＝max(α，u_om). Wherein u_omSaturated defeated for systemEnter value, $\max (\mathrm{\&alpha;},u_{\mathrm{om}})=\left\{\begin{array}{cc}\mathrm{\&alpha;}& \mathrm{\&alpha;}>u_{\mathrm{om}}\\ u_{\mathrm{om}}& \mathrm{\&alpha;}\&le;u_{\mathrm{om}}\end{array}.\right.$

The present invention's advantage is compared with prior art:

A kind of sliding-mode control with anti-interference fault freedom of the present invention adopts sliding mode controller and fault to examineDisconnected observer and interference observer combination, be applicable to input the anti-interference fault-tolerant control of saturation system. Fault is examinedDisconnected observer and interference observer have ensured the fault-tolerant interference free performance of system, and fault and interference are estimatedAnd counteracting, sliding mode controller has robustness for disturbance, and the full of system considered in the design of sliding formwork gainAnd input value. Control method has solved the fault-tolerant anti-interference problem of input saturation system. Meanwhile, two of designsObserver is offset the fault in system and interference, has weakened the chattering phenomenon of sliding formwork control, improves appearanceThe precision of sliding-mode control and reliability in state control system.

Brief description of the drawings

Fig. 1 is the design flow diagram of a kind of sliding-mode control with anti-interference fault freedom of the present invention.

Detailed description of the invention

As shown in Figure 1, specific implementation step of the present invention following (taking three axis stabilized satellite attitude control system asExample is carried out the specific implementation of illustration method):

1, consider that fault and multi-source in satellite attitude control system disturb, and build system dynamics model

Eulerian angles between three axis stabilized satellite body coordinate system and orbital coordinate system are very little, and the attitude of satellite is movingMechanics and kinematics model linearisation can obtain:

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

Three equations of above formula are followed successively by satellite roll axle, pitch axis, three axial attitude dynamicses of yaw axisEquation. J₁，J₂，J₃Be respectively three axle rotary inertias, φ, θ, ψ is respectively satellite body coordinate system and orbit coordinateThree axle Eulerian angles between system;Be respectively three axle Eulerian angles speed;Be respectively three axle Eulerian anglesAcceleration; u₁，u₂，u₃Be respectively three axle control moments; ω₀For satellite orbit angular speed; F (t) for time accident barrier,T_d1，T_d2，T_d3Be respectively the disturbance torque (comprising the disturbance torque that sensor and executing agency bring) of three axles;

Following steps have the sliding formwork of anti-interference fault freedom with satellite pitch channel kinetic model for example designController, roll axle is identical with yaw axis method for designing.

Expectation attitude angle, angular speed and the angular acceleration of three axis stabilized satellite are designated as respectively θ_c(t)、WithAnd be zero. Can obtain error system equation 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_c(t)=θ (t) is error angle,For error angle acceleration.

Write pitch channel attitude error model as 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, multi-source EVAC state variableU (t) is control inputs,F (t) for time accident barrier, d₁(t) be modeling to disturb, d₂(t) be modeling to disturb, d₁And d (t)₂(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 disturbance d₁(t) by following external disturbance model ∑₁Represent:

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

Wherein, w (t) be can modeling interference model state variable, V be can modeling interference model outputMatrix, W represents system battle array that can modeling interference model, δ (t) is the can not modeling random (of energy boundedL₂NormBounded) disturb B₃For the gain battle array of can not modeling disturbing.

2, design error failure diagnostic observations device and interference observer are distinguished suspected fault and can modeling be disturbed

Hindering F (t) design error failure diagnostic observations device for the time accident in system is:

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

For the d that can modeling disturbs in system₁(t) design interference observer is:

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

Wherein,For the estimated value of fault,For Interference Estimation value,For the estimated value of w (t), ξ (t)With r (t) is respectively the auxiliary variable in fault diagnosis observer and interference observer, K and L are respectively and treatFixed fault diagnosis observer gain matrix and interference observer gain matrix, tried to achieve by subsequent step 3.

Failure definition evaluated error $e_F\left(t\right)=F\left(t\right)-\hat{F}\left(t\right),$ Interference Estimation error $e_w\left(t\right)=w\left(t\right)-\hat{w}\left(t\right);$

According to the expression formula of fault diagnosis observer, can obtain Fault Estimation error equation and be:

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

According to the expression formula of interference observer, can obtain Interference Estimation error equation and be:

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

3, fault diagnosis observer gain matrix and interference observer gain matrix solve

The evaluated error equation can modeling disturbing in connection row second step and the evaluated error equation of fault asUnder:

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

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 isUnit matrix, z_∞(t) be H_∞Performance is with reference to output, and C is H_∞Performance output matrix.

Utilize convex optimized algorithm to solve the fault-tolerant anti-interference controller gain battle array of multi-source EVAC; At the beginning of givenInitial value e_wAnd e (0)_F(0), output matrix C, disturbs inhibition degree γ₁、γ₂And γ₃, solve following protruding optimization and askTopic:

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 above formula, symbol * represents the symmetry blocks of appropriate section in symmetrical matrix, sym (PW₁+RB₁E) expression formulaAs follows: sym (PW₁+RB₁E)＝(PW₁+RB₁E)+(PW₁+RB₁E)^T。

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

4, design sliding mode controller, the estimated value of operational failure and interference is compensate for failed and interference respectively

1) switching function design is as follows:

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

Wherein, e_θ(t)＝θ(t)-θ_c(t), three axis stabilized satellite is expected attitude angle θ_c(t)=0, therefore have

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

2) sliding mode design of control law is as follows:

u(t)＝u_eq(t)+u_vs(t)

Wherein, u_eq(t) be controlled quentity controlled variable of equal value, u_vs(t) be switch controlled quentity controlled variable.

OrderHaveThe kinetics equation of substitution pitch axis can obtain

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

Use can modeling interference and the estimated value of faultReplace respectively actual value d₁(t)、F(t)，Try to achieve $u_{\mathrm{eq}}\left(t\right)=3{{\mathrm{\&omega;}}_0}^2(J_1-J_3)\mathrm{\&theta;}\left(t\right)-J_2{k}_1\stackrel{.}{\mathrm{\&theta;}}\left(t\right)-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right).$

Switch controlled quentity controlled variable is designed to u_vs(t)＝-T_pSgn (s (t)). Wherein T_pFor sliding formwork gain, by subsequent step5 try to achieve, and sgn (s (t)) is switch function, represent by following form:

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

System control inputs is 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, solve sliding formwork gain, ensure system stability

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

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

Control inputs is expressed to formula substitution above formula, can 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 theorem, whenSet up, proof system can reach sliding-mode surface, and slidingDynamic model state plane is asymptotically stable;

Note α=|| b₁e_F(t)+b₁Ve_w(t)+b₂d₂(t) ||. Obviously, need to meet T_p>=α, hasSystem can reach sliding-mode surface, and reaches asymptotic consistent stable state.

Consider the saturated input problem of system, T_p＝max(α，u_om). Wherein u_omKnown, for flywheel is carriedThe maximum input torque of confession, $\max (\mathrm{\&alpha;},u_{\mathrm{om}})=\left\{\begin{array}{cc}\mathrm{\&alpha;}& \mathrm{\&alpha;}>u_{\mathrm{om}}\\ u_{\mathrm{om}}& \mathrm{\&alpha;}\&le;u_{\mathrm{om}}\end{array}\right..$

For avoiding the chattering phenomenon of sliding mode controller output, adopt saturation function sat (s (t)) to replace switch letterNumber sgn (s (t)). Sat (s (t)) expression formula 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, σ is the factor of quivering that disappears, and object is both effectively to eliminate and buffet, and ensures again system Fast Convergent,Value is in (0.02,0.08) scope.

It is known existing that the content not being described in detail in description of the present invention belongs to professional and technical personnel in the fieldTechnology.

Claims (1)

1. there is a sliding-mode control for anti-interference fault freedom, it is characterized in that comprising the following steps:First, set up the kinetic model of system; Secondly, design error failure diagnostic observations device and interference observer are estimatedFault in meter systems and can modeling disturbing; Again, solve interference observer and fault diagnosis observerGain matrix; Then, design sliding mode controller, will disturb with Fault Estimation value substitution sliding mode controller inCompensation is of equal value to be disturbed and fault; Finally, input under saturated prerequisite and solve sliding formwork gain in system, ensureSystem stability; Concrete steps are as follows:

The first step, sets up system dynamics model

Build and comprise the system dynamics model disturbing with fault, as follows:

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

Wherein, x₁(t),x₂(t),…,x_n(t) be system mode, n >=2 are positive integer, and u (t) is control inputs, F (t)For the time accident of rate of change bounded hinders, d₁(t) be modeling to disturb, d₂(t) be can not modeling random disturbances,a₀、a₁、…a_n-1With b₁、b₂Be internal system parameter; d₁(t) by following external disturbance model Σ₁Represent:

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

Wherein, w (t) be can modeling interference model state variable, V be can modeling interference model output matrix,W represents system battle array that can modeling interference model, B₃For gain battle array that can not modeling random disturbances, δ (t) isEnergy bounded can not modeling random disturbances;

Selecting system state variable X (t)=[x₁(t)x₂(t)……x_n(t)]^T, write as state-space expression asUnder:

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

Wherein, X (t) is system state variables, and A is system battle array, B₁For input matrix, B₂For can not modelingThe gain battle array of random disturbances;

Second step, design error failure diagnostic observations device and interference observer are distinguished suspected fault and can modeling be disturbed

Hindering F (t) design error failure diagnostic observations device for the time accident in system is:

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

For the d that can modeling disturbs in system₁(t) design interference observer is:

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

Wherein,For the estimated value of fault,For Interference Estimation value,For the estimated value of w (t),ξ (t) and r (t) are respectively the auxiliary variable in fault diagnosis observer and interference observer, and K and L are respectivelyFault diagnosis observer gain matrix undetermined and interference observer gain matrix, tried to achieve by follow-up the 3rd step;

Failure definition evaluated error isDisturbance-observer error is $e_w\left(t\right)=w\left(t\right)-\hat{w}\left(t\right);$

Can obtain Fault Estimation error equation according to the expression formula of fault diagnosis observer is:

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

Can obtain Interference Estimation error equation according to the expression formula of interference observer as follows:

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

The 3rd step, fault diagnosis observer gain matrix and interference observer gain matrix solve

Interference Estimation error equation and Fault Estimation error equation in connection row second step 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 isUnit matrix, z_∞(t) be H_∞Performance is with reference to output, and C is H_∞Performance is adjustable output matrix;

Utilize convex optimized algorithm solve multi-source EVAC can modeling interference observer gain matrix and fault examineDisconnected observer gain matrix; Given initial value e_wAnd e (0)_F(0), adjustable output matrix C, disturbs inhibition degreeγ₁、γ₂And γ₃, solve following protruding 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, symbol * represents the symmetry blocks of appropriate section in symmetrical matrix, sym (PW₁+RB₁E) expression formula asUnder:

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

Above formula is solved to obtain to P, R, observer gain matrix $\left[\begin{array}{c}L\\ K\end{array}\right]=P^{-1}R;$

The 4th step, design sliding mode controller, the estimated value of operational failure and interference is compensate for failed and interference respectivelyThe design procedure of sliding mode controller is as follows:

1) design sliding-mode surface s (t)

The method for designing of sliding-mode surface is as follows:

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

2) design sliding formwork control law

Adopt function switching law, comprise input of equal value and switch input two parts, input of equal value byTry to achieve; Design of control law is as follows:

u(t)＝u_eq(t)+u_vs(t)

Wherein u_eq(t) be the controlled quentity controlled variable of equal value of system, u_vs(t) be switch controlled quentity controlled variable;

OrderHaveThe kinetic model of substitution system can obtain

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

The u (t) being obtained by above formula is controlled quentity controlled variable of equal value, and then can obtain:

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

Use can modeling interference and the estimated value of faultReplace respectively actual value d₁(t)、F(t)，Have $u_{eq}\left(t\right)=1/b_1(\underset{i=1}{\overset{n}{\&Sigma;}}{a}_{n-i}{x}_i\left(t\right)-\underset{i=2}{\overset{n}{\&Sigma;}}{k}_{i-1}{x}_i\left(t\right))-{\hat{d}}_1\left(t\right)-\hat{F}\left(t\right);$

Switch controlled quentity controlled variable is designed to u_vs(t)＝-T_pSgn (s (t)); Wherein, T_pFor sliding formwork gain, by the 5th stepTry to achieve; Sgn (s (t)) is switch function, represents by following form:

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

Control inputs expression formula is:

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

The 5th step, solves sliding formwork gain, ensures system stability

Liapunov function is designed to

By the definition of s (t) and the kinetic model of system, can obtain

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

The control inputs that the 4th step is tried to achieve is expressed formula substitution above formula, has

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

According to Lyapunov theorem, whenSet up, proof system can reach sliding formworkFace, and sliding mode plane is asymptotically stable;

Note α=|| b₁e_F(t)+b₁Ve_w(t)+b₂d₂(t) ||, obviously, only need to meet T_p>=α, hasSystem can reach sliding-mode surface, and reaches asymptotic consistent stable state;

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