CN104049537A - Non-affine non-linear flight control system robust adaptive fault-tolerant control system - Google Patents

Abstract

The invention discloses a non-affine non-linear flight control system robust adaptive fault-tolerant control system used for a non-linear system with parameters in a non-affine mode. An observer has relatively idea robustness on the situation that the parameters change in a large range. Fault information and disturbance information are implied in the observer, and then a fault-tolerant controller is dynamically designed on the basis of the observer. Due to the non-affine non-linear system, the controller is not easy to design, the non-affine non-linear system approximates to an affine non-linear system with time-varying parameters, and the parameters needing to be known are estimated online through a filter. By means of the non-linear flight control system, effectiveness of a method is verified, and robust adaptive fault-tolerant control of the non-affine non-linear flight control system can be achieved. Robust adaptive fault-tolerant control of the non-affine non-linear flight control system is achieved and applied in the flight control system, and a simulation result displays effectiveness of the method.

Description

The non-linear flight control system robust adaptive of nonaffine fault-tolerant control system

Technical field

The invention belongs to flight control system technical field, relate in particular to the non-linear flight control system robust adaptive of a kind of nonaffine fault-tolerant control system.

Background technology

At present, the nonlinear Control based on model has obtained significant progress in theoretical and application, as feedback linearization, and sliding formwork control, inverting control etc.Document [155-156] has done corresponding summary and conclusion to more current main nonlinear control methods.Adaptive technique, due to energy On-line Estimation unknown parameter, is therefore used for designing fault-tolerant control with many nonlinear control method combinations widely.And adaptive control to need estimative parameter and control inputs be affine form, uncertain parameter and control inputs must be dominant form, or with the pass of state variable be linearization relation.In flight control system, conventional method is near linearization trim point, if when the quantity of state that aircraft is current and control inputs present nonaffine form, linearizing model become while being exactly, so the controller based near inearized model design trim point may cause closed-loop system unstable, even system is dispersed.

The fault-tolerant controller that designs a non-affine nonlinear systems is not a simple thing, has two difficult points to solve fully, the one, how to design an adaptive parameter estimation algorithm, and the 2nd, how to design a Reconfigurable Control algorithm.A more common adaptive parameter estimation algorithm is exactly by near system model Taylor series expansion parameter and standard value, utilizes the low order item design parameter observer of Taylor series.The system perturbing among a small circle for parameter like this can obtain good estimation, and for the system of this class parameter wide variation of fault, such method is difficult to obtain desirable estimates of parameters, if there is external disturbance in system simultaneously, can there is again error in the parameter of estimating, even do not realize the estimation of parameter.So how must inquire into for the parameter estimator design load of the nonaffine Nonlinear Uncertain Systems design ideal under fault.All there is certain deficiency in the capable of reconstructing controller of more existing non-affine nonlinear systems, conventional method of inverse needs the contrary of searching system model, although document [156] has proved a controllable system and has certainly existed the contrary of it, but looking for an inverse system is not that suggestion is easy to thing, as control inputs lies in sine and cosine functions.Document [137] proposes a kind of nonaffine controller design method, but the shortcoming of the method maximum is exactly the exponent number of meeting increase system.The method that document [157] separates based on markers has designed a kind of nonaffine controller, is difficult to and existing adaptive technique the effectively combination such as sliding mode technology but the method weak point is exactly the method.In order to provide a kind of effectively nonaffine controller design method, author proposes a kind of controller design method in document [158], introduces a wave filter for approximate evaluation linearization working point.

The achievement in research that the fault-tolerant control of non-affine nonlinear systems is relevant is at present little, known to author, only have Song Yongduan professor within 2011, on robotization journal, to write the one section of paper [159] aspect this, but the method is only for SISO system, does not relate to temporarily mimo system.

Summary of the invention

The object of the embodiment of the present invention is to provide the non-linear flight control system robust adaptive of a kind of nonaffine fault-tolerant control system, is intended to solve the relevant little problem of achievement in research of fault-tolerant control of current non-affine nonlinear systems.

The embodiment of the present invention is to realize like this, the non-linear flight control system robust adaptive of a kind of nonaffine fault-tolerant control system, the non-linear flight control system robust adaptive of this nonaffine fault-tolerant control system comprises: reference model, controller, the non-linear controlled device of nonaffine, wave filter, backup system;

Reference model connects controller, and controller connects the non-linear controlled device of nonaffine and wave filter, and backup system connects controller and the non-linear controlled device of nonaffine.

Further, the non-linear flight control system robust adaptive of this nonaffine fault-tolerant control system is considered following non-affine nonlinear systems:

$\stackrel{\·}{x}=f(x,u)+d\left(t\right)---\left(5.1\right)$

Wherein: x ∈ R ^rfor state vector, u ∈ R ⁿfor input vector, d ∈ R ^rfor the external disturbance vector of unknown bounded, f (.) is nonlinear function, is represented by the fault model obtaining after each input channel Actuators Failures:

$\begin{array}{c}{u}_i={\mathrm{\&sigma;}}_i{u}_{\mathrm{ci}},{\mathrm{\&sigma;}}_i\&Element;[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]\\ 0<{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i\&le;1,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i\&GreaterEqual;1,i=\mathrm{1,2},...,n\end{array}---\left(5.2\right)$

Wherein σ _ifor the unknown inefficacy factor, for the known inefficacy factor sigma of definition _imaximin, work as σ _i=1 represents that non-fault occurs, so control inputs exists Actuators Failures representation for fault to be:

u(t)＝[u ₁(t)，u ₂(t)，……，u _n(t)] ^T＝∑u _c(t) (5.3)

Wherein ∑=diag (σ ₁..., σ _n), so the non-affine nonlinear systems (5.1) under fault is expressed as:

$\stackrel{\&CenterDot;}{x}=f(x,{\mathrm{\&Sigma;u}}_c)+d\left(t\right)---(5.4)$

Equation (5.4) is write as general type:

$\stackrel{\&CenterDot;}{x}=f(x,u_c\mathrm{\&sigma;})+d\left(t\right)---(5.5)$

Wherein σ=[σ ₁..., σ _n] ^t, be convenient to carrying out of work, provide a hypothesis below;

Suppose 1:f (x, u _c, σ) and be x, u _c, the smooth continuous derivatived functions of σ, and control inputs u _cbounded, the output reference model of system (5.1) is:

${\stackrel{\&CenterDot;}{x}}_n=A_m{x}_m+B_{m}r---\left(5.6\right)$

Wherein: x _m∈ R ^rfor the state vector of reference model, A _mbe a stable model reference system matrix, r ∈ R ^lfor the input of reference model;

The object of robust Fault-Tolerant Control designs fault-tolerant control inputs u exactly _c(t), in the situation that there is external disturbance and Actuators Failures fault, guarantee || x (t)-x _m(t) ||≤ε;

Can be found out by (5.5), the fault parameter of the non-affine nonlinear systems under actuator failures and control inputs variable do not show and are contained in function.

Further, the implementation method of backup system comprises:

Definition for the estimated value of σ, by hypothesis 1, by function f (x, u _c, σ) near carry out single order Taylor series expansion, obtain:

$f(x,u_c,\mathrm{\&sigma;})=f(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})(\mathrm{\&sigma;}-\widehat{\mathrm{\&sigma;}})+\mathrm{\&xi;}\left(t\right)---\left(5.7\right)$

Wherein:

$g_1(x,u_c,\widehat{\mathrm{\&sigma;}})=\frac{\&PartialD;f(x,u_c,\mathrm{\&sigma;})}{\&PartialD;\mathrm{\&sigma;}}{|}_{\mathrm{\&sigma;}=\widehat{\mathrm{\&sigma;}}},\mathrm{\&xi;}\left(t\right)=\underset{i=2}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\&PartialD;}^{i}f(x,u_c,\mathrm{\&sigma;})}{\&PartialD;{\mathrm{\&sigma;}}^i}{|}_{\mathrm{\&sigma;}=\widehat{\mathrm{\&sigma;}}}{(\mathrm{\&sigma;}-\widehat{\mathrm{\&sigma;}})}^i---\left(5.8\right)$

Based on (5.7) and (5.8), (5.5) are write again as following equation:

$\stackrel{.}{x}=f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{\&sigma;}+\mathrm{\&upsi;}\left(t\right)---\left(5.9\right)$

Wherein:

$f_1(x,u_c,\widehat{\mathrm{\&sigma;}})=f(x,u_c,\widehat{\mathrm{\&sigma;}})-g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}---\left(5.10\right)$

υ(t)＝ξ(t)+d(t) (5.11)

Find out that υ (t) is unknown and bounded, is defined as

Definition ε=z-x, wherein z is the observed reading of state x, for (5.9), observer is as follows:

$\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right)---\left(5.12\right)$

And drawn by following adaptive law

$\stackrel{.}{\widehat{\mathrm{\&sigma;}}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}---\left(5.13\right)$

Wherein γ ₁> 0, P=P ^t> 0 and P are A ^tthe solution of P+PA=-Q, wherein Q=Q ^t> 0, A is a Hurwitz matrix, guarantees the minimum value of estimated value in setting and maximal value between, the design of sliding formwork item is as follows;

$\stackrel{.}{\widehat{\mathrm{\&sigma;}}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}---\left(5.14\right)$

Time-varying parameter m (t) is upgraded and is obtained by following adaptive law:

$\stackrel{.}{m}\left(t\right)=\mathrm{\&Gamma;}{\mathrm{\&epsiv;}}^T\mathrm{\&epsiv;},m\left(0\right)>0---\left(5.15\right)$

Definition inefficacy factor evaluated error is by observer equation (5.12) and equation (5.9), obtain observational error dynamic equation and be:

$\stackrel{.}{\widehat{\mathrm{\&sigma;}}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}---\left(5.16\right).$

Further, by observer adaptive updates rule $\stackrel{.}{\widehat{\mathrm{\&sigma;}}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}$ With sliding formwork item $\stackrel{.}{\widehat{\mathrm{\&sigma;}}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\},$ Observational error dynamic equation asymptotically stable in the large,, to arbitrary initial value ε (0), guarantees lim _{t → ∞}ε (t)=0, damage Fault Estimation error bounded;

Serialization sliding formwork item is as follows:

$v\left(t\right)=-\frac{\mathrm{P\&epsiv;}}{\left|\right|\mathrm{P\&epsiv;}\left|\right|+\mathrm{\&rho;}}m\left(t\right)---\left(5.17\right)$

Wherein: ρ=ρ ₀+ ρ ₁|| ε ||, and ρ ₀and ρ ₁for being greater than 0 constant.

Further, the implementation method of controller and stability analysis comprises:

Based on observer utilize the non-affine nonlinear systems controller implementation method of putting forward early stage, first definition observer $\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right)$ Write as following obtaining:

$\stackrel{.}{z}=\mathrm{A\&epsiv;}+F(x,u_c,\widehat{\mathrm{\&sigma;}})+v\left(t\right)---\left(5.18\right)$

Choose u _nat u _cnear, and will at u _nplace carries out Taylor series expansion and obtains:

$F(x,u_c,\widehat{\mathrm{\&sigma;}})=F(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})(u_c-u_n)+O\left(t\right)---\left(5.19\right)$

Wherein:

$F_d(x,u_n,\widehat{\mathrm{\&sigma;}})=\frac{\&PartialD;F(x,u_c,\widehat{\mathrm{\&sigma;}})}{\&PartialD;u}{|}_{u_c=u_n},O\left(t\right)=\underset{i=2}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\&PartialD;}^{i}F(x,u_c,\widehat{\mathrm{\&sigma;}})}{\&PartialD;u^i}{|}_{u_c=u_n}{(u_c-u_n)}^i---\left(5.20\right)$

Definition $F_n(x,u_n,\widehat{\mathrm{\&sigma;}})=F(x,u_n,\widehat{\mathrm{\&sigma;}})-F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_n,$ (5.18) are expressed as again:

$\stackrel{.}{z}=\mathrm{A\&epsiv;}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_c+v\left(t\right)+O\left(t\right)---\left(5.21\right)$

Found out by (5.19), if u _nmore approach u _c, the higher order indefinite small O (t) of Taylor series more trends towards 0, $\underset{u_n\&RightArrow;u_c}{\lim }O\left(t\right)=0---\left(5.22\right)$

Due to u in reality _cbe that the controller being designed calculates, current time is unknown, so cannot directly obtain near the u it _nso, introduce wave filter here for estimating and definite u _n, the wave filter of introducing is as follows:

${\stackrel{.}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c---\left(5.23\right)$

Therefore by wave filter (5.23), obtain lim _{ζ → ∞}u _n=u _c, i.e. lim _{ζ → ∞}o (t)=0, so by above analysis, observer dynamic equation (5.18) is expressed as:

$\begin{array}{c}{\stackrel{.}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ \stackrel{.}{\hat{x}}=A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_c+v\left(t\right)+O\left(t\right)\end{array}---\left(5.24\right)$

Definition observer state variable tracking error be utilize dynamic inverse, as follows based on equation (5.24) design control law:

$\begin{array}{c}{\stackrel{.}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ u_c=-F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})[K\hat{e}+A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+v\left(t\right)-A_m{x}_m-B_{m}r]\end{array}---\left(5.25\right)$

Ride gain K can be obtained by following Riccati Solving Equations:

K ^TP ₁+P ₁K＝-Q ₁ (5.26)

Wherein $P_1=P_1^T>0,Q_1=Q_1^T>0.$

Further, define system tracking error e=x-x _m, failure system at controller $\begin{array}{c}{\stackrel{.}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ u_c=-F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})[K\hat{e}+A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+v\left(t\right)-A_m{x}_m-B_{m}r]\end{array},$ And observer $\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right),\stackrel{.}{\widehat{\mathrm{\&sigma;}}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_1,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_1]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}$ under the closed-loop system of composition, can ensure system asymptotic tracking reference locus, i.e. lim _{ζ → ∞, t → ∞}e=0;

Prove: will in control law (5.25) substitution (5.24), obtain observer error dynamics equation:

$\stackrel{\&CenterDot;}{\hat{e}}=K\hat{e}+O\left(t\right)---\left(5.27\right)$

Select following Lyapunov equation:

$V_1={\hat{e}}^T{P}_1\hat{e}---\left(5.28\right)$

To V ₁differentiate, and utilize Young inequality 2ab≤ε a ^ta+ ε ^-1b ^tb obtains:

Wherein:

λ _min(.), λ _max(.) is minimax eigenvalue matrix, therefore uses the consistent final bounded lemma of the overall situation, obtains V ₁exponential convergence, and finally can converge to following territory:

Because lim _{ζ → ∞}o (t)=0, can obtain again by the result of theorem so be easy to obtain lim _{ζ → ∞, t → ∞}e (t)=0.

Further, contrary may existence, for avoiding this kind of situation to occur, often adopt in practice following formula to replace that is:

$F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})=F_d^T(x,u_n,\widehat{\mathrm{\&sigma;}}){[F_d(x,u_n,\widehat{\mathrm{\&sigma;}})F_d^T(x,u_n,\widehat{\mathrm{\&sigma;}})+\mathrm{\&alpha;}]}^{-1}$

Wherein α is positive definite matrix.

The non-linear flight control system robust adaptive of nonaffine provided by the invention fault-tolerant control system, has the nonlinear system of nonaffine form, and exists observer in wide variation situation still can have quite desirable robustness in parameter for parameter; Observer all implies failure message and disturbance information wherein, then based on observer dynamic design fault-tolerant controller, because system is non-affine nonlinear systems, the design of controller is also not easy, non-affine nonlinear systems is approximately to an Affine Incentive nonlinear system with time-varying parameter, and the required parameter of knowing is carried out On-line Estimation by a wave filter.Utilize a nonaffine flight control system to verify the validity of institute's extracting method, can realize the robust Fault-Tolerant Control of non-affine nonlinear systems.The present invention has realized the fault-tolerant control of robust adaptive of non-affine nonlinear systems, and is applied in flight control system, and simulation result shows the validity of institute's extracting method.

Brief description of the drawings

Fig. 1 is the structural representation of the non-linear flight control system robust adaptive of the nonaffine fault-tolerant control system that provides of the embodiment of the present invention;

In figure: 1, reference model; 2, controller; 3, the non-linear controlled device of nonaffine; 4, wave filter; 5, backup system;

Fig. 2 is the system responses curve synoptic diagram in the situation 1 that provides of the embodiment of the present invention;

Fig. 3 is the system responses curve synoptic diagram in the situation 2 that provides of the embodiment of the present invention;

Fig. 4 is the system responses curve synoptic diagram in the situation 3 that provides of the embodiment of the present invention;

Fig. 5 is estimates of parameters and the sliding formwork item response schematic diagram in the situation 3 that provides of the embodiment of the present invention.

Embodiment

In order to make object of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, the present invention is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not intended to limit the present invention.

Below in conjunction with drawings and the specific embodiments, application principle of the present invention is further described.

As shown in Figure 1, the non-linear flight control system robust adaptive of the nonaffine of embodiment of the present invention fault-tolerant control system mainly by: reference model 1, controller 2, the non-linear controlled device 3 of nonaffine, wave filter 4, backup system 5 form;

Reference model 1 connects controller 2, and controller 2 connects the non-linear controlled device 3 of nonaffine and wave filter 4, and backup system 5 connects controller 2 and the non-linear controlled device 3 of nonaffine;

Specific embodiments of the invention:

1, problem is described:

Consider following non-affine nonlinear systems:

$\stackrel{\&CenterDot;}{x}=f(x,u)+d\left(t\right)---\left(5.1\right)$

Wherein: x ∈ R ^rfor state vector, u ∈ R ⁿfor input vector, d ∈ R ^rfor the external disturbance vector of unknown bounded, f (.) is nonlinear function, in the present invention, using Actuators Failures as research situation, do not considering in Actuator dynamic situation, the fault model that can be obtained after each input channel Actuators Failures by previous section can represent:

$\begin{array}{c}{u}_i={\mathrm{\&sigma;}}_i{u}_{\mathrm{ci}},{\mathrm{\&sigma;}}_i\&Element;[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]\\ 0<{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i\&le;1,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i\&GreaterEqual;1,i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,n\end{array}---\left(5.2\right)$

Wherein σ _ifor the unknown inefficacy factor, for the known inefficacy factor sigma of definition _imaximin, work as σ _i=1 represents that non-fault occurs, so control inputs exists Actuators Failures fault to be expressed as:

u(t)＝[u ₁(t)，u ₂(t)，……，u _n(t)] ^T＝∑u _c(t) (5.3)

Wherein ∑=diag (σ ₁..., σ _n), so the non-affine nonlinear systems (5.1) under fault can be expressed as:

$\stackrel{\&CenterDot;}{x}=f(x,\mathrm{\&Sigma;}{u}_c)+d\left(t\right)---\left(5.4\right)$

Equation (5.4) can be write as general type:

$\stackrel{\&CenterDot;}{x}=f(x,u_c,\mathrm{\&sigma;})+d\left(t\right)---(5.5)$

Wherein σ=[σ ₁..., σ _n] ^t, be convenient to carrying out of work, provide a hypothesis below;

Suppose 1:f (x, u _c, σ) and be x, u _c, the smooth continuous derivatived functions of σ, and control inputs u _cbounded, the output reference model of system (5.1) is:

${\stackrel{\&CenterDot;}{x}}_m=A_m{x}_m+B_{m}r---\left(5.6\right)$

Wherein: x _m∈ R ^rfor the state vector of reference model, A _mbe a stable model reference system matrix, r ∈ R ^lfor the input of reference model;

The object of robust Fault-Tolerant Control designs fault-tolerant control inputs u exactly _c(t), in the situation that there is external disturbance and Actuators Failures fault, guarantee || x (t)-x _m(t) ||≤ε;

Can be found out by (5.5), the fault parameter of the non-affine nonlinear systems under actuator failures and control inputs variable do not show and are contained in function, this brings very large difficulty to the estimation of fault parameter and the design of controller, the present invention provides based on adaptive sliding mode Design of Observer parameter estimation algorithm, and provide a kind of new nonaffine Design of non-linear controllers method, design a kind of fault-tolerant controller of non-affine nonlinear systems;

2, fault-tolerant control system:

2.1, backup system:

Definition for the estimated value of σ, by hypothesis 1, by function f (x, u _c, σ) near carry out single order Taylor series

Launch, can obtain:

$f(x,u_c,\mathrm{\&sigma;})=f(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})(\mathrm{\&sigma;}-\widehat{\mathrm{\&sigma;}})+\mathrm{\&xi;}\left(t\right)---\left(5.7\right)$

Wherein:

$g_1(x,u_c,\widehat{\mathrm{\&sigma;}})=\frac{\&PartialD;f(x,u_c,\mathrm{\&sigma;})}{\&PartialD;\mathrm{\&sigma;}}{|}_{\mathrm{\&sigma;}=\widehat{\mathrm{\&sigma;}}},\mathrm{\&xi;}\left(t\right)=\underset{i=2}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\&PartialD;}^{i}f(x,u_c,\mathrm{\&sigma;})}{{\&PartialD;\mathrm{\&sigma;}}^i}{|}_{\mathrm{\&sigma;}=\widehat{\mathrm{\&sigma;}}}{(\mathrm{\&sigma;}-\widehat{\mathrm{\&sigma;}})}^i---\left(5.8\right)$

Based on (5.7) and (5.8), (5.5) can be write again as following equation:

$\stackrel{\&CenterDot;}{x}=f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{\&sigma;}+\mathrm{\&upsi;}\left(t\right)---\left(5.9\right)$

Wherein:

$f_1(x,u_c,\widehat{\mathrm{\&sigma;}})=f(x,u_c,\widehat{\mathrm{\&sigma;}})-g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}---\left(5.10\right)$

υ(t)＝ξ(t)+d(t) (5.11)

Can find out that υ (t) is unknown and bounded, is defined as

Definition ε=z-x, wherein z is the observed reading of state x, for (5.9), designs following observer:

$\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right)---\left(5.12\right)$

And drawn by following adaptive law

$\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&sigma;}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}---\left(5.13\right)$

Wherein γ ₁> 0, P=P ^t> 0 and P are A ^tthe solution of P+PA=-Q, wherein Q=Q ^t> 0, A is a Hurwitz matrix, it can guarantee the minimum value of estimated value in setting and maximal value between, the design of sliding formwork item is as follows;

$v\left(t\right)=\left\{\begin{array}{cc}-\frac{\mathrm{P\&epsiv;}}{\left|\right|\mathrm{P\&epsiv;}\left|\right|}m\left(t\right)& \mathrm{if}\left|\right|\mathrm{P\&epsiv;}\left|\right|\&NotEqual;0\\ 0& \mathrm{otherwise}\end{array}\right.---\left(5.14\right)$

Time-varying parameter m (t) is upgraded and is obtained by following adaptive law:

$\stackrel{\&CenterDot;}{m}\left(t\right)=\mathrm{\&Gamma;}{\mathrm{\&epsiv;}}^T\mathrm{\&epsiv;},m\left(0\right)>0---\left(5.15\right)$

Definition inefficacy factor evaluated error is by observer equation (5.12) and equation (5.9), can obtain observational error dynamic equation and be:

$\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}=\mathrm{A\&epsiv;}+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widetilde{\mathrm{\&sigma;}}+v\left(t\right)-\mathrm{\&upsi;}\left(t\right)---\left(5.16\right)$

Theorem 5.1: by observer (5.12), adaptive updates rule (5.13) and sliding formwork item (5.14), can observational error dynamic equation (5.16) asymptotically stable in the large, to arbitrary initial value ε (0), guarantee lim _{t → ∞}ε (t)=0, damage Fault Estimation error bounded;

Prove: the similar theorem 3.1 of proof procedure;

Card is finished.

Serialization sliding formwork item is as follows:

$v\left(t\right)=-\frac{\mathrm{P\&epsiv;}}{\left|\right|\mathrm{P\&epsiv;}\left|\right|+\mathrm{\&rho;}}m\left(t\right)---\left(5.17\right)$

Wherein: ρ=ρ ₀+ ρ ₁|| ε ||, and ρ ₀and ρ ₁for being greater than 0 constant;

Remarks 5.1: the present invention exists near dynamic equation (5.5) is carried out to single order Taylor series expansion, here (parameter changes greatly the approximate inaccurate problem of model of avoiding parameter to change causing greatly, the higher order term that Taylor series approximation is ignored will be larger, at this moment higher order term just can not be left in the basket), and the method that the present invention carries, because approximate model is a parameter time varying affine nonlinear system, and as long as parameter estimation algorithm can have convergence faster than observer in the observer of design, just can ensure the accuracy of approximate model;

2.2, controller design and stability analysis:

Based on observer (5.12), utilize the non-affine nonlinear systems controller design method of carrying early stage, first definition observer (5.12) can be write as following obtaining:

$\stackrel{\&CenterDot;}{z}=\mathrm{A\&epsiv;}+F(x,u_c,\widehat{\mathrm{\&sigma;}})+v\left(t\right)---\left(5.18\right)$

Choose u _nat u _cnear, and will at u _nplace carries out Taylor series expansion and obtains:

$F(x,u_c,\widehat{\mathrm{\&sigma;}})=F(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})(u_c-u_n)+O\left(t\right)---\left(5.19\right)$

Wherein:

$F_d(x,u_n,\widehat{\mathrm{\&sigma;}})=\frac{\&PartialD;F(x,u_c,\widehat{\mathrm{\&sigma;}})}{\&PartialD;u}{|}_{u_c=u_n},O\left(t\right)=\underset{i=2}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\&PartialD;}^{i}F(x,u_c,\widehat{\mathrm{\&sigma;}})}{{\&PartialD;u}^i}{|}_{u_c=u_n}{(u_c-u_n)}^i---\left(5.20\right)$

Definition $F_n(x,u_n,\widehat{\mathrm{\&sigma;}})=F(x,u_m,\widehat{\mathrm{\&sigma;}})-F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_n,$ (5.18) can be expressed as again:

$\stackrel{\&CenterDot;}{z}=\mathrm{A\&epsiv;}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_c+v\left(t\right)+O\left(t\right)---\left(5.12\right)$

Can be found out by (5.19), if u _nmore approach u _c, the higher order indefinite small O (t) of Taylor series gets over

To in 0, $\underset{u_n\&RightArrow;u_c}{\lim }O\left(t\right)=0---\left(5.22\right)$

Due to u in reality _cbe that the controller being designed calculates, current time is unknown, so cannot directly obtain near the u it _nso, introduce wave filter here for estimating and definite u _n, the wave filter of introducing is as follows:

${\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c---\left(5.23\right)$

Therefore by wave filter (5.23), can obtain lim _{ζ → ∞}u _n=u _c, i.e. lim _{ζ → ∞}o (t)=0, so by above analysis, observer dynamic equation (5.18) can be expressed as:

$\begin{array}{c}{\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ \stackrel{\stackrel{\&CenterDot;}{^}}{x}=A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_c+v\left(t\right)+O\left(t\right)\end{array}---\left(5.24\right)$

Definition observer state variable tracking error be utilize dynamic inverse, as follows based on equation (5.24) design control law:

$\begin{array}{c}{\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ u_c=-F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})[K\hat{e}+A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+v\left(t\right)-A_m{x}_m-B_{m}r]\end{array}---\left(5.25\right)$

Ride gain K can be obtained by following Riccati Solving Equations:

K ^TP ₁+P ₁K＝-Q ₁ (5.26)

Wherein $P_1=P_1^T>0,Q_1=Q_1^T>0;$

Theorem 5.2: define system tracking error e=x-x _m, failure system (5.5) is in controller (5.25), and under the closed-loop system of observer (5.12)-(5.14) composition, can ensure system asymptotic tracking reference locus, i.e. lim _{ζ → ∞, t → ∞}e=0;

Prove: will in control law (5.25) substitution (5.24), can obtain observer error dynamics equation:

$\stackrel{\stackrel{\&CenterDot;}{^}}{e}=K\hat{e}+O\left(t\right)---\left(5.27\right)$

Select following Lyapunov equation:

$V_1={\hat{e}}^T{P}_1\hat{e}---\left(5.28\right)$

To V ₁differentiate, and utilize Young inequality 2ab≤ε a ^ta+ ε ^-1b ^tb obtains:

Wherein:

λ _min(.), λ _max(.) is minimax eigenvalue matrix, therefore uses the consistent final bounded lemma of the overall situation, can obtain V ₁exponential convergence, and finally can converge to following territory:

Because lim _{ζ → ∞}o (t)=0, can obtain again by the result of theorem 5.1 so be easy to obtain lim _{ζ → ∞, t → ∞}e (t)=0;

Card is finished.

Remarks 5.2: contrary may existence, for avoiding this kind of situation to occur, often adopt in practice following formula to replace that is:

$F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})=F_d^T(x,u_n,\widehat{\mathrm{\&sigma;}}){[F_d(x,u_n,\widehat{\mathrm{\&sigma;}})F_d^T(x,u_n,\widehat{\mathrm{\&sigma;}})+\mathrm{\&alpha;}]}^{-1}---\left(5.32\right)$

Wherein α is positive definite matrix;

For convenience of reader understanding, provide design frame chart of the present invention here as Fig. 1.

By following simulating, verifying, effect of the present invention is further described:

Next, utilize document ^[137,160]provide the validity that unmanned aerial vehicle flight path angle and speed control system carry out emulation proof institute extracting method;

Dynamic model is:

$\begin{array}{c}\stackrel{\&CenterDot;}{V}=g(\frac{T-D}{W}-\sin \mathrm{\&gamma;})\\ \stackrel{\&CenterDot;}{\mathrm{\&gamma;}}=\frac{g}{V}(n\cos \mathrm{\&mu;}-\cos \mathrm{\&gamma;})\\ \stackrel{\&CenterDot;}{\mathrm{\&chi;}}=\frac{\mathrm{gn}\sin \mathrm{\&mu;}}{V\cos \mathrm{\&gamma;}}\end{array}---\left(5.33\right)$

Definition flying speed V, path angle γ, position angle, path χ is state variable, thrust T, load factor n, and pitch angle μ is control inputs, resistance D computing formula is as follows:

$D=0.5\mathrm{\&rho;}{V}^{2}S{C}_{D_0}+\frac{2{\mathrm{kn}}^2{W}^2}{\mathrm{\&rho;}{V}^{2}S}---\left(5.34\right)$

Model parameter is as shown in table 1:

Table 1 model parameter value

Definition status variable x=[V, γ, χ] ^t, control inputs is u=[T, n, μ] ^t, the Actuators Failures factor is σ=[σ ₁, σ ₂, σ ₃] ^t, external disturbance is d=[d ₁, d ₂, d ₃] ^t=[0.2cos (2t), 0.0002sin (t), 0.0002cos (t)] ^t, equation (5.33) can be expressed as:

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=c_{11}{x}_1^2+\frac{c_{12}{\mathrm{\&sigma;}}_2^2{u}_2^2}{x_1^2}+c_{13}\sin \left(x_2\right)+c_{14}{\mathrm{\&sigma;}}_1{u}_1+d_1\\ {\stackrel{\&CenterDot;}{x}}_2=\frac{1}{x_1}(c_{21}\cos \left(x_2\right)+c_{22}{\mathrm{\&sigma;}}_2{u}_2\cos \left({\mathrm{\&sigma;}}_3{u}_3\right))+d_2\\ {\stackrel{\&CenterDot;}{x}}_3=\frac{c_{31}{\mathrm{\&sigma;}}_2{u}_2\sin \left({\mathrm{\&sigma;}}_3{u}_3\right)}{x_1\cos \left(x_2\right)}+d_3\end{array}---\left(5.35\right)$

Wherein: c ₁₂=-2kgW/ (ρ S), c ₁₃=-g, c ₁₄=g/W, c ₂₁=-g, c ₂₂=g, c ₃₁=g, the reference locus of setting speed v is 300m/s, path angle γ, the reference locus of position angle, path χ is exported by two following reference models:

$\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_{m1}=x_{m2}\\ {\stackrel{\&CenterDot;}{x}}_{m2}=-{9x}_{m1}-{6x}_{m2}+{9r}_{\mathrm{\&gamma;}}\left(t\right)\end{array}---\left(5.35\right)$

${\stackrel{\&CenterDot;}{x}}_m=-x_m+r_{\mathrm{\&chi;}}\left(t\right)---\left(5.36\right)$

Wherein r _χ(t)=30sin (π t/18) deg;

$r_{\mathrm{\&gamma;}}\left(t\right)=\left\{\begin{array}{ccc}0& \mathrm{deg}& t\&le;15\\ 0.5(t-15)& \mathrm{deg}& 15<t\&le;25\\ 5& \mathrm{deg}& 25<t\&le;35\\ 0.5(45-t)& \mathrm{deg}& 35<t\&le;45\\ 0& \mathrm{deg}& t>45\end{array}\right.---\left(5.37\right)$

Suppose that Actuators Failures fault occurs as follows:

$\begin{array}{c}{\mathrm{\&sigma;}}_1=\left\{\begin{array}{cc}1.5(2.6/3-0.02t)& 10<t\&le;30\\ 1.5(0.02t-1/3)& 30<t\&le;50\\ 1& \mathrm{others}\end{array}\right.\\ {\mathrm{\&sigma;}}_2=\left\{\begin{array}{cc}1& t\&le;20\\ 0.6& 20<t\end{array}\right.\\ {\mathrm{\&sigma;}}_3=\left\{\begin{array}{cc}1& t\&le;20\\ 0.8& 20<t\end{array}\right.\end{array}---\left(5.38\right)$

State initial value is V (0)=300m/s, γ (0)=0deg, χ (0)=0deg, m (0)=0.15, σ (0)=[1,1,1] ^t, design aiding system parameter is A=diag (2 ,-2 ,-2), γ ₁=2, Г=1000, ρ ₀=5, designing filter parameter is ζ=50, controller gain K=diag (1,1,1), P=diag (0.3,1800,2000);

Situation 1: consider under normal circumstances, the control of designed controller to non-affine nonlinear systems, controller is designed to:

$\begin{array}{c}{\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ u_c=-F_d^{-1}[F_n{|}_{\widehat{\mathrm{\&sigma;}}=1}+\mathrm{Ke}-A_m{x}_m-B_{m}r]\end{array}---\left(5.39\right)$

As shown in Figure 2, the nonaffine control method carried is as seen from Figure 2 effectively to the response curve of system tracking error, can realize preferably and estimate to follow the tracks of.

Situation 2: after fault (5.38) occurs, what controller still used is (5.39), system keeps track mistake

Poor response curve is shown in Fig. 3, can find out in the situation of not carrying out fault-tolerant control, can not realize the input track reference track of system;

Situation 3: after fault (5.38) occurs, adopt the fault-tolerant controller of the present invention's design, system tracking error response curve is shown in Fig. 4, the estimation of parameter σ with sliding formwork item v (t) response curve as shown in Figure 5, can find out that the present invention proposes fault-tolerant control and can make system under failure condition, still can ensure that system has good tracking performance.

The present invention is directed to exist and disturb and the uncertain non-affine nonlinear systems of parameter, under given Fault-tolerant Control System Design framework, provide a kind of nonaffine Nonlinear Fault Tolerant controller design method, designed observer can be applicable to parameter and have the nonlinear system of nonaffine form, and exist observer in wide variation situation still can have quite desirable robustness in parameter, observer all implies failure message and disturbance information wherein, then based on observer dynamic design fault-tolerant controller, because system is non-affine nonlinear systems, the design of controller is also not easy, the present invention provides a kind of dynamically non-affine nonlinear systems approximation method, non-affine nonlinear systems is approximately to an Affine Incentive nonlinear system with time-varying parameter, and the required parameter of knowing is carried out On-line Estimation by a wave filter, utilize a nonaffine flight control system to verify the validity of institute's extracting method, can realize the robust Fault-Tolerant Control of non-affine nonlinear systems.

The foregoing is only preferred embodiment of the present invention, not in order to limit the present invention, all any amendments of doing within the spirit and principles in the present invention, be equal to and replace and improvement etc., within all should being included in protection scope of the present invention.

Claims (7)

1. the non-linear flight control system robust adaptive of a nonaffine fault-tolerant control system, it is characterized in that, the non-linear flight control system robust adaptive of this nonaffine fault-tolerant control system comprises: reference model, controller, the non-linear controlled device of nonaffine, wave filter, backup system;

Reference model connects controller, and controller connects the non-linear controlled device of nonaffine and wave filter, and backup system connects controller and the non-linear controlled device of nonaffine.

2. the non-linear flight control system robust adaptive of nonaffine as claimed in claim 1 fault-tolerant control system, is characterized in that, the non-linear flight control system robust adaptive of this nonaffine fault-tolerant control system is considered following non-affine nonlinear systems:

$\stackrel{\&CenterDot;}{x}=f(x,u)+d\left(t\right)---\left(5.1\right)$

Wherein: x ∈ R ^rfor state vector, u ∈ R ⁿfor input vector, d ∈ R ^rfor the external disturbance vector of unknown bounded, f (.) is nonlinear function, is represented by the fault model obtaining after each input channel Actuators Failures:

$\begin{array}{c}{u}_i={\mathrm{\&sigma;}}_i{u}_{\mathrm{ci}},{\mathrm{\&sigma;}}_i\&Element;[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]\\ 0<{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i\&le;1,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i\&GreaterEqual;1,i=\mathrm{1,2},...,n\end{array}---\left(5.2\right)$

Wherein σ _ifor the unknown inefficacy factor, for the known inefficacy factor sigma of definition _imaximin, work as σ _i=1 represents that non-fault occurs, so control inputs exists Actuators Failures representation for fault to be:

u(t)＝[u ₁(t)，u ₂(t)，……，u _n(t)] ^T＝∑u _c(t) (5.3)

Wherein ∑=diag (σ ₁..., σ _n), so the non-affine nonlinear systems (5.1) under fault is expressed as:

$\stackrel{\&CenterDot;}{x}=f(x,{\mathrm{\&Sigma;u}}_c)+d\left(t\right)---\left(5.4\right)$

Equation (5.4) is write as general type:

$\stackrel{\&CenterDot;}{x}=f(x,u_c,\mathrm{\&sigma;})+d\left(t\right)---\left(5.5\right)$

Wherein σ=[σ ₁..., σ _n] ^t, be convenient to carrying out of work, provide a hypothesis below;

Suppose 1:f (x, u _c, σ) and be x, u _c, the smooth continuous derivatived functions of σ, and control inputs u _cbounded, the output reference model of system (5.1) is:

${\stackrel{\&CenterDot;}{x}}_m=A_m{x}_m+B_{m}r---\left(5.6\right)$

Wherein: x _m∈ R ^rfor the state vector of reference model, A _mbe a stable model reference system matrix, r ∈ R ^lfor the input of reference model;

The object of robust Fault-Tolerant Control designs fault-tolerant control inputs u exactly _c(t), in the situation that there is external disturbance and Actuators Failures fault, guarantee || x (t)-x _m(t) ||≤ε;

Can be found out by (5.5), the fault parameter of the non-affine nonlinear systems under actuator failures and control inputs variable do not show and are contained in function.

3. the non-linear flight control system robust adaptive of nonaffine as claimed in claim 1 or 2 fault-tolerant control system, is characterized in that, the implementation method of backup system comprises:

Definition for the estimated value of σ, by hypothesis 1, by function f (x, u _c, σ) near carry out single order Taylor series expansion, obtain:

$f(x,u_c,\mathrm{\&sigma;})=f(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})(\mathrm{\&sigma;}-\widehat{\mathrm{\&sigma;}})+\mathrm{\&xi;}\left(t\right)---\left(5.7\right)$

Wherein:

$g_1(x,u_c,\widehat{\mathrm{\&sigma;}})=\frac{\&PartialD;f(x,u_c,\mathrm{\&sigma;})}{\&PartialD;\mathrm{\&sigma;}}{|}_{\mathrm{\&sigma;}=\widehat{\mathrm{\&sigma;}}},\mathrm{\&xi;}\left(t\right)=\underset{i=2}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\&PartialD;}^{i}f(x,u_c,\mathrm{\&sigma;})}{{\&PartialD;\mathrm{\&sigma;}}^i}{|}_{\mathrm{\&sigma;}=\widehat{\mathrm{\&sigma;}}}{(\mathrm{\&sigma;}-\widehat{\mathrm{\&sigma;}})}^i---\left(5.8\right)$

Based on (5.7) and (5.8), (5.5) are write again as following equation:

$\stackrel{\&CenterDot;}{x}=f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{\&sigma;}+\mathrm{\&upsi;}\left(t\right)---\left(5.9\right)$

Wherein:

$f_1(x,u_c,\widehat{\mathrm{\&sigma;}})=f(x,u_c,\widehat{\mathrm{\&sigma;}})-g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}---\left(5.10\right)$

υ(t)＝ξ(t)+d(t) (5.11)

Find out that υ (t) is unknown and bounded, is defined as

Definition ε=z-x, wherein z is the observed reading of state x, for (5.9), observer is as follows:

$\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right)---\left(5.12\right)$

And drawn by following adaptive law

$\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&sigma;}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}---\left(5.13\right)$

Wherein γ ₁> 0, P=P ^t> 0 and P are A ^tthe solution of P+PA=-Q, wherein Q=Q ^t> 0, A is a Hurwitz matrix, guarantees the minimum value of estimated value in setting σ _iand maximal value between, the design of sliding formwork item is as follows;

$\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&sigma;}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}---\left(5.14\right)$

Time-varying parameter m (t) is upgraded and is obtained by following adaptive law:

$\stackrel{\&CenterDot;}{m}\left(t\right)={\mathrm{\&Gamma;\&epsiv;}}^T\mathrm{\&epsiv;},m\left(0\right)>0---\left(5.15\right)$

Definition inefficacy factor evaluated error is by observer equation (5.12) and equation (5.9), obtain observational error dynamic equation and be:

$\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&sigma;}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}---\left(5.16\right).$

4. the non-linear flight control system robust adaptive of nonaffine as claimed in claim 3 fault-tolerant control system, is characterized in that, by observer $\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right),$ Adaptive updates rule $\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&sigma;}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}$ With sliding formwork item $\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&sigma;}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\},$ Observational error dynamic equation asymptotically stable in the large,, to arbitrary initial value ε (0), guarantees lim _{t → ∞}ε (t)=0, damage Fault Estimation error bounded;

Serialization sliding formwork item is as follows:

$v\left(t\right)=-\frac{\mathrm{P\&epsiv;}}{\left|\right|\mathrm{P\&epsiv;}\left|\right|+\mathrm{\&rho;}}m\left(t\right)---\left(5.17\right)$

Wherein: ρ=ρ ₀+ ρ ₁|| ε ||, and ρ ₀and ρ ₁for being greater than 0 constant.

5. the non-linear flight control system robust adaptive of nonaffine as claimed in claim 1 fault-tolerant control system, is characterized in that, the implementation method of controller and stability analysis comprises:

Based on observer utilize the non-affine nonlinear systems controller implementation method of putting forward early stage, first definition observer $\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right)$ Write as following obtaining:

$\stackrel{\&CenterDot;}{z}=\mathrm{A\&epsiv;}+F(x,u_c,\widehat{\mathrm{\&sigma;}})+v\left(t\right)---\left(5.18\right)$

Choose u _nat u _cnear, and will at u _nplace carries out Taylor series expansion and obtains:

$F(x,u_c,\widehat{\mathrm{\&sigma;}})=F(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})(u_c-u_n)+O\left(t\right)---\left(5.19\right)$

Wherein:

$F_d(x,u_n,\widehat{\mathrm{\&sigma;}})=\frac{\&PartialD;F(x,u_c,\widehat{\mathrm{\&sigma;}})}{\&PartialD;u}{|}_{u_c=u_n},O\left(t\right)=\underset{i=2}{\overset{\&infin;}{\mathrm{\&Sigma;}}}\frac{{\&PartialD;}^{i}F(x,u_c,\widehat{\mathrm{\&sigma;}})}{\&PartialD;u^i}{|}_{u_c=u_n}{(u_c-u_n)}^i---\left(5.20\right)$

Definition $F_n(x,u_n,\widehat{\mathrm{\&sigma;}})=F(x,u_n,\widehat{\mathrm{\&sigma;}})-F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_n,$ (5.18) are expressed as again:

$\stackrel{\&CenterDot;}{z}=\mathrm{A\&epsiv;}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_c+v\left(t\right)+O\left(t\right)---\left(5.21\right)$

Found out by (5.19), if u _nmore approach u _c, the higher order indefinite small O (t) of Taylor series more trends towards 0, $\underset{u_n\&RightArrow;u_c}{\lim }O\left(t\right)=0---\left(5.22\right)$

Due to u in reality _cbe that the controller being designed calculates, current time is unknown, so cannot directly obtain near the u it _nso, introduce wave filter here for estimating and definite u _n, the wave filter of introducing is as follows:

${\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c---\left(5.23\right)$

Therefore by wave filter (5.23), obtain lim _{ζ → ∞}u _n=u _c, i.e. lim _{ζ → ∞}o (t)=0, so by above analysis, observer dynamic equation (5.18) is expressed as:

$\begin{array}{c}{\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ \stackrel{\stackrel{\&CenterDot;}{^}}{x}=A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+F_d(x,u_n,\widehat{\mathrm{\&sigma;}})u_c+v\left(t\right)+O\left(t\right)\end{array}---\left(5.24\right)$

Definition observer state variable tracking error be utilize dynamic inverse, as follows based on equation (5.24) design control law:

$\begin{array}{c}{\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ u_c=-F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})[K\hat{e}+A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+v\left(t\right)-A_m{x}_m-B_{m}r]\end{array}---\left(5.25\right)$

Ride gain K can be obtained by following Riccati Solving Equations:

K ^TP ₁+P ₁K＝-Q ₁ (5.26)

Wherein $P_1=P_1^T>0,Q_1=Q_1^T>0.$

6. the non-linear flight control system robust adaptive of nonaffine as claimed in claim 5 fault-tolerant control system,

It is characterized in that define system tracking error e=x-x _m, failure system at controller

$\begin{array}{c}{\stackrel{\&CenterDot;}{u}}_n=-\mathrm{\&zeta;}{u}_n+\mathrm{\&zeta;}{u}_c\\ u_c=-F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})[K\hat{e}+A\tilde{x}+F_n(x,u_n,\widehat{\mathrm{\&sigma;}})+v\left(t\right)-A_m{x}_m-B_{m}r]\end{array},$ And observer

$\hat{z}=A(z-x)+f_1(x,u_c,\widehat{\mathrm{\&sigma;}})+g_1(x,u_c,\widehat{\mathrm{\&sigma;}})\widehat{\mathrm{\&sigma;}}+v\left(t\right),\stackrel{\stackrel{\&CenterDot;}{^}}{\mathrm{\&sigma;}}={\mathrm{Proj}}_{[{\underset{\&OverBar;}{\mathrm{\&sigma;}}}_i,{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}_i]}\{-2{\mathrm{\&gamma;}}_1{g_1}^T(x,u_c,\widehat{\mathrm{\&sigma;}})\mathrm{P\&epsiv;}\}$

under the closed-loop system of composition, can ensure system asymptotic tracking reference locus, i.e. lim _{ζ → ∞, t → ∞}e=0;

Prove: will in control law (5.25) substitution (5.24), obtain observer error dynamics equation:

$\stackrel{\stackrel{\&CenterDot;}{^}}{e}=K\hat{e}+O\left(t\right)---\left(5.27\right)$

Select following Lyapunov equation:

$V_1={\hat{e}}^T{P}_1\hat{e}---\left(5.28\right)$

To V ₁differentiate, and utilize Young inequality 2ab≤ε a ^ta+ ε ^-1b ^tb obtains:

Wherein:

λ _min(.), λ _max(.) is minimax eigenvalue matrix, therefore uses the consistent final bounded lemma of the overall situation, obtains V ₁exponential convergence, and finally can converge to following territory:

Because lim _{ζ → ∞}o (t)=0, can obtain again by the result of theorem so be easy to obtain lim _{ζ → ∞, t → ∞}e (t)=0.

7. the non-linear flight control system robust adaptive of nonaffine as claimed in claim 5 fault-tolerant control system, is characterized in that, contrary may existence, for avoiding this kind of situation to occur, often adopt in practice following formula to replace that is:

$F_d^{-1}(x,u_n,\widehat{\mathrm{\&sigma;}})=F_d^T(x,u_n,\widehat{\mathrm{\&sigma;}}){[F_d(x,u_n,\widehat{\mathrm{\&sigma;}})F_d^T(x,u_n,\widehat{\mathrm{\&sigma;}})+\mathrm{\&alpha;}]}^{-1}$

Wherein α is positive definite matrix.