Interval error observer-based aircraft engine active fault tolerant control method

Abstract

The present invention provides an interval error observer-based aircraft engine active fault tolerant control method, and belongs to the technical field of aircraft control. The method comprises: tracking the state and the output of a reference model of an aircraft engine through an error feedback controller; compensating a control system of the aircraft engine having a disturbance signal and actuator and sensor faults through a virtual sensor and a virtual actuator; observing an error between a system with fault of the aircraft engine and the reference model through an interval error observer, and feeding back the error to the error feedback controller; and finally, using a difference between the output of the reference model of the system with fault and the output of the virtual actuator as a control signal to realize active fault tolerant control of the aircraft engine.

Claims

1. An interval error observer-based aircraft engine active fault tolerant control method for a controller of an aircraft engine, comprising the following steps: step 1.1: establishing an affine parameter-dependent aircraft engine linear-parameter-varying LPV model
{dot over (x)}.sub.p(t)=[A.sub.0+ΔA(θ)]x.sub.p(t)+[B.sub.0+ΔB(θ)]u.sub.p(t)+d.sub.f(t)
y.sub.p(t)=C.sub.px.sub.p(t)+v(t)  (1) where R.sup.m and R.sup.m×n respectively represent a m-dimensional real number column vector and a m-row n-column real matrix; state vectors x.sub.p=[Y.sub.nl Y.sub.nh].sup.T ∈ R.sup.n.sup.x, Y.sub.nl and Y.sub.nh respectively represent variation of relative conversion speed of low pressure and high pressure rotors; n.sub.x represents the dimension of a state variable x; n.sub.y represents the dimension of an output vector y; n.sub.u represents the dimension of control input u.sub.p; control input u.sub.p=U.sub.p.sub.f ∈ R.sup.n.sup.x is a fuel pressure step signal; output vectors y.sub.p=Y.sub.nh ∈ R.sup.n.sup.y, A.sub.0 ∈ R.sup.n.sup.x.sup.×n.sup.x, B.sub.0 ∈ R.sup.n.sup.x.sup.×n.sup.x and C.sub.p ∈ R.sup.n.sup.y.sup.×n.sup.x are known system constant matrices; d.sub.f(t) is a disturbance variable; the relative conversion speed n.sub.h of the high pressure rotor of the aircraft engine is a scheduling parameter θ ∈ R.sup.p; system variable matrices ΔA(θ) and ΔB(θ) satisfy −ΔA≤ΔA(θ)≤ΔA and −ΔB≤ΔB(θ)≤ΔB; ΔA ∈ R.sup.n.sup.x.sup.×n.sup.x is an upper bound of ΔA(θ); ΔB ∈ R.sup.n.sup.x.sup.×n.sup.u is an upper bound of ΔB(θ); ΔA≥0, ΔB≥0; a state variable initial value x.sub.p(0) satisfies x.sub.0≤x.sub.p(0)≤x.sub.0; x.sub.0, x.sub.0 ∈ R.sup.n.sup.x are respectively known upper bound and lower bound of the state variable initial value x.sub.p(0); d,d ∈ R.sup.n.sup.x are known upper bound and lower bound of an unknown disturbance d.sub.f(t); sensor noise v(t) satisfies |v(t)|<V; V is a known bound; V>0; step 1.2: defining reference model of fault-free system of the aircraft engine (1) as
{dot over (x)}.sub.pref(t)=A.sub.0x.sub.pref(t)+B.sub.0u.sub.pref(t)
y.sub.pref(t)=C.sub.px.sub.pref(t)  (2) where x.sub.pref ∈ R.sup.n.sup.x is a reference state vector of the fault-free system; u.sub.pref ∈ R.sup.n.sup.x is control input of the fault-free system; y.sub.pref ∈ R.sup.n.sup.y is a reference output vector; an error feedback controller of the fault-free system of the aircraft engine is designed according to the aircraft engine LPV model established in the step 1.1; step 1.2.1: defining an error e.sub.p(t)=x.sub.pref(t)−x.sub.p(t) between the affine parameter-dependent aircraft engine LPV model and the reference model of the fault-free system of the aircraft engine to obtain error state equations of the fault-free system:
ė.sub.p(t)=[A.sub.0+ΔA(θ)]e.sub.p(t)+[B.sub.0+ΔB(θ)]Δu.sub.cp(t)−ΔA(θ)x.sub.pref(t)−ΔB(θ)u.sub.pref(t)−d.sub.f(t)
ε.sub.cp(t)=C.sub.pe.sub.p(t)−v(t)  (3) where Δu.sub.cp(t) and ε.sub.cp(t) represent the input and output difference between the reference model and aircraft engine LPV model with Δu.sub.cp(t)=u.sub.pref(t)−u.sub.p(t) and ε.sub.cp(t)=y.sub.pref(t)−y.sub.p(t), respectively; step 1.2.2: representing state equations of the upper bound ē.sub.p and the lower bound e.sub.p of the error vector e.sub.p as:
{dot over (ē)}.sub.p(t)=[A.sub.0−LC.sub.p]ē.sub.p(t)+[B.sub.0+ΔB]Δu.sub.cp(t)+Lε.sub.cp(t)+|L|V−d(t)+ΔA|x.sub.pref(t)|+ϕ.sub.p(t)
{dot over (e)}.sub.p(t)=[A.sub.0−LC.sub.p]e.sub.p(t)+[B.sub.0−ΔB]Δu.sub.cp(t)+Lε.sub.cp(t)−|L|V−d(t)−ΔA|x.sub.pref(t)|−ϕ.sub.p(t)  (4) where ē.sub.p, e.sub.p ∈ R.sup.n.sup.x are respectively the upper bound and the lower bound of the error vector e.sub.p, i.e., e.sub.p(t)≤e.sub.p(t)≤ē.sub.p(t); ϕ.sub.p(t)=ΔA(ē.sub.p.sup.+(t)+e.sub.p.sup.−(t)), ē.sub.p.sup.+=max {0, ē.sub.p}, ē.sub.p.sup.−=ē.sub.p.sup.+−ē.sub.p, e.sub.P.sup.+=max {0, ē.sub.p}, e.sub.p.sup.−=e.sub.p.sup.+−e.sub.p; L ∈ R.sup.n.sup.x.sup.×n.sup.y is an error gain matrix of the fault-free system and satisfies A.sub.0−LC.sub.p ∈ M.sup.n.sup.x.sup.×n.sup.x; M.sup.n.sup.x represents a set of n.sub.x-dimensional Metzler matrix; |L| represents taking absolute values of all elements of the matrix L; step 1.2.3: respectively setting e.sub.pa=0.5(ē.sub.p+e.sub.p) and e.sub.pd=ē.sub.p−e.sub.p, which represent the middle value and range of the interval of e.sub.p, respectively; rewriting the formula (4) as:
ė.sub.pd(t)=[A.sub.0−LC.sub.p]e.sub.pd(t)+2ΔBΔu.sub.cp(t)+ϕ.sub.pd(t)+δ.sub.pd(t)
ė.sub.pa(t)=[A.sub.0−LC.sub.p]e.sub.pa(t)+B.sub.0Δu.sub.cp(t)+LC.sub.pe.sub.p(t)+δ.sub.pa(t)  (5) where ϕ.sub.pd(t), δ.sub.pa(t) and δ.sub.pd(t) are variables defined as
ϕ.sub.pd(t)=2ΔA(ē.sub.p.sup.−(t)+e.sub.p.sup.−(t))
δ.sub.pd(t)=2|L|V−d(t)+d(t)+2ΔA|x.sub.pref(t)|
δ.sub.pa(t)=−Lv(t)−0.5(d(t)+d(t))  (6) step 1.2.4: defining output signal of the error feedback controller as:
Δu.sub.cp(t)=K.sub.ae.sub.pa(t)+K.sub.de.sub.pd(t)  (7) where K.sub.d, K.sub.a ∈ R.sup.n.sup.x.sup.×n.sup.x represent gain matrices of the error feedback controller signal (7); setting e.sub.x(t)=e.sub.p(t)−e.sub.pa(t), −0.5e.sub.pd(t)≤e.sub.x(t)≤0.5e.sub.pd(t), and then
ė.sub.pa(t)=[A.sub.0+B.sub.0K.sub.a]e.sub.pa(t)+B.sub.0K.sub.de.sub.pd(t)+LC.sub.pe.sub.x(t)+δ.sub.pa(t)  (8) step 1.2.5: rewriting formulas (5) and (8) as: $\begin{array}{cc}{\dot{\xi }}_p\left(t\right)=G_p\left(t\right){\xi }_p\left(t\right)+{\delta }_p\left(t\right)& \left(9\right)\end{array}$ $\begin{array}{cc}{G}_p\left(t\right)=\left[\begin{array}{cc}{A}_0-L{C}_p& 0\\ B_0{K}_d& A_0+B_0{K}_a\end{array}\right]+A_{pd}\left(t\right)& \left(10\right)\end{array}$ where ξ.sub.p(t) is an error vector composed of the range of the error interval e.sub.pd and middle value of the error interval e.sub.pa with ${\xi }_p\left(t\right)={\left[{e_{pd}\left(t\right)}^T,{e_{pa}\left(t\right)}^T\right]}^T,{\delta }_p\left(t\right)={\left[{\left({\delta }_{pd}\left(t\right)+2\stackrel{\_}{\Delta B}\Delta u_{cp}\left(t\right)\right)}^T,{{\delta }_{pa}\left(t\right)}^T\right]}^T$ and then $\begin{array}{cc}\left[\begin{array}{c}{\varphi }_{pd}\\ L{C}_p{e}_x\end{array}\right]=A_{pd}\left[\begin{array}{c}{e}_{pd}\\ e_{pa}\end{array}\right]& \left(11\right)\end{array}$ step 1.2.6: S.sup.m×m representing an m-dimensional real symmetric square matrix; setting a matrix E,F ∈ SR.sup.2n.sup.x.sup.×2n.sup.x; E,F 0 representing that each element in E,F is greater than 0; constant λ>0; and obtaining a matrix inequality:
G.sub.p.sup.TE+EG.sub.p+λE+F0  (12) namely, setting each element in G.sub.p.sup.TE+EG.sub.p+λE+F to be less than 0; solving the matrix inequality (12) to obtain the gain matrices K.sub.d, K.sub.a of the error feedback controller so as to obtain the error feedback controller signal (7); step 1.3: describing the aircraft engine LPV model having disturbance and actuator and sensor faults as:
{dot over (x)}.sub.f(t)=[A.sub.0+ΔA(θ)]x.sub.f(t)+B.sub.f(γ(t))u.sub.f(t)+d.sub.f(t)
y.sub.f(t)=C.sub.f(ϕ(t))x.sub.f+v(t)  (13) where x.sub.f ∈ R.sup.n.sup.x is a state vector of a system with fault; u.sub.f ∈ R.sup.n.sup.x is the control input of the system with fault; y.sub.f ∈ R.sup.n.sup.y is an output vector of the system with fault; B.sub.f(γ(t)) and C.sub.f(ϕ(t)) are respectively actuator and sensor faults, expressed as
B.sub.f(γ(t))=[B.sub.0+ΔB(θ)]diag(γ.sub.1(t), . . . ,γ.sub.n(t))
C.sub.f(ϕ(t))=C.sub.p diag(ϕ.sub.1(t), . . . ,ϕ.sub.n(t))  (14) where 0≤y.sub.i(t)≤1 and 0≤ϕ.sub.j(t)≤1 respectively represent the failure degree of the i th actuator and the j th sensor; γ.sub.i=1 and γ.sub.i=0 respectively represent health and complete failure of the i th actuator; ϕ.sub.j is similar; diag(γ.sub.1, γ.sub.2, . . . , γ.sub.n) represents a diagonal matrix with diagonal elements γ.sub.1, γ.sub.2, . . . , γ.sub.n; diag(ϕ.sub.1, ϕ.sub.2, . . . , ϕ.sub.n) is similar; setting γ(t) and ϕ(t) estimated values respectively as {circumflex over (γ)}(t) and {circumflex over (ϕ)}(t), and then
B.sub.f(γ(t))=B.sub.f({circumflex over (γ)}(t))+B.sub.f(Δγ(t))
C.sub.f(ϕ(t))=C.sub.f(ϕ(t))+C.sub.f(Δϕ(t))  (15) where Δγ(t)=γ(t)−{circumflex over (γ)}(t) and Δϕ(t)=ϕ(t)−{circumflex over (ϕ)}(t) are respectively errors of estimation of γ(t) and ϕ(t); a virtual actuator and a virtual sensor are respectively designed according to the actuator and sensor faults; step 1.3.1: designing the virtual sensor as:
{dot over (x)}.sub.vs(t)=A.sub.vs(θ)x.sub.vs(t)+B.sub.f({circumflex over (γ)}(t))Δu(t)+Qy.sub.f(t)
{circumflex over (γ)}.sub.f(t)=C.sub.vsx.sub.vs(t)+Py.sub.f(t)  (16)
where
A.sub.vs(θ)=A.sub.0+ΔA(θ)−QC.sub.f({circumflex over (ϕ)}(t))
C.sub.vs=C.sub.p−PC.sub.f({circumflex over (ϕ)}(t))  (17) where x.sub.vs ∈ R.sup.n.sup.x is a state variable of a virtual sensor; Δu ∈ R.sup.n.sup.x is a difference in control inputs of a fault model and a fault reference model; {circumflex over (γ)}.sub.f ∈ R.sup.n.sup.y is an output vector of the virtual sensor; Q and P are respectively parameter matrices of the virtual sensor; step 1.3.2: a linear matrix inequality (LMI) region S.sub.1(ρ.sub.1, q.sub.1, r.sub.1, θ.sub.1) representing an intersection of a left half complex plane region with a bound of −ρ.sub.1, a circular region with a radius of r.sub.1 and a circle center of q.sub.1 and a fan region having an intersection angle θ.sub.1 with a negative real axis; representing a state matrix A.sub.vs of the virtual sensor as a polytope structure; A.sub.vsj=A.sub.0+ΔA(θ.sub.j)−Q.sub.jC.sub.f({circumflex over (ϕ)}(t)), where θ.sub.j represents the value of the j th vertex θ; A.sub.vsj represents the value of the state matrix A.sub.vs of the virtual sensor of the j th vertex; a necessary and sufficient condition for eigenvalues of A.sub.vsj to be in S.sub.1(ρ.sub.1, q.sub.1, r.sub.1, θ.sub.1) is that there exists a symmetrical matrix X.sub.1>0 so that the linear matrix inequalities (18)-(20) are established, thereby obtaining a parameter matrix Q.sub.j of the virtual sensor of the corresponding vertex; $\begin{array}{cc}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1+{X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T+2{\rho }_1{X}_1<0& \left(18\right)\end{array}$ $\begin{array}{cc}\left[\begin{array}{cc}-r_1{X}_1& q_1{X}_1+\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\end{array}\right]X_1\\ q_1{X}_1+{X_1\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\end{array}\right]}^T& -r_1{X}_1\end{array}\right]<0& \left(19\right)\end{array}$ $\begin{array}{cc}\left(\begin{array}{c}\sin {\theta }_1\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1+\\ {X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T\end{array}\right\}\\ \cos {\theta }_1\left\{\begin{array}{c}{X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T-\\ \left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1\end{array}\right\}\end{array}\begin{array}{c}\cos {\theta }_1\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1-\\ {X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T\end{array}\right\}\\ \sin {\theta }_1\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1+\\ {X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T\end{array}\right\}\end{array}\right)<0& \left(20\right)\end{array}$ selecting Q.sub.j of a vertex corresponding to θ.sub.j as a parameter matrix of the virtual sensor; step 1.3.3: representing the parameter matrix P of the virtual sensor as:
P=C.sub.pC.sub.f.sup.†  (21) where † represents pseudo-inversion of the matrix; step 1.3.4: designing the virtual actuator as
{dot over (x)}.sub.va(t)=A.sub.vax.sub.va(t)+B.sub.vaΔu.sub.c(t)
Δu(t)=Mx.sub.va(t)+NΔu.sub.c(t)
y.sub.c(t)=ŷ.sub.f(t)+C.sub.px.sub.va(t)  (22)
where
A.sub.va=A.sub.0+ΔA(θ)−B.sub.f({circumflex over (γ)}(t))M
B.sub.va=B.sub.0+ΔB(θ)−B.sub.f({circumflex over (γ)}(t))N  (23) where x.sub.va ∈ R.sup.n.sup.x is a state variable of the virtual actuator; Δu.sub.c ∈ R.sup.n.sup.x is the output of the error feedback controller; y.sub.c ∈ R.sup.n.sup.y is an output vector of the virtual actuator; M and N are respectively parameter matrices of the virtual actuator; step 1.3.5: a linear matrix inequality (LMI) region S.sub.2(ρ.sub.2, q.sub.2, r.sub.2, θ.sub.2) representing an intersection of a left half complex plane region with a bound of −ρ.sub.2, a circular region with a radius of r.sub.2 and a circle center of q.sub.2 and a fan region having an intersection angle θ.sub.2 with a negative real axis; representing a state matrix A.sub.va of the virtual actuator as a polytope structure; A.sub.vaj=A.sub.0+ΔA(θ.sub.j)−B.sub.f({circumflex over (γ)}(t))M.sub.j, where θ.sub.j represents the value of the j th vertex θ; A.sub.vaj represents the value of the state matrix A.sub.va of the virtual actuator of the j th vertex; a necessary and sufficient condition for eigenvalues of A.sub.vaj to be in S.sub.2(ρ.sub.2, q.sub.2, r.sub.2, θ.sub.2) is that there exists a symmetrical matrix X.sub.2>0 so that the linear matrix inequalities (24)-(26) are established, thereby obtaining a parameter matrix M.sub.i of the virtual actuator; $\begin{array}{cc}\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2+{X_2\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T+2{\rho }_2{X}_2<0& \left(24\right)\end{array}$ $\begin{array}{cc}\left[\begin{array}{cc}-r_2{X}_2& q_2{X}_2+\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ B_f\left(\hat{\gamma }\left(t\right)\right)M_i\end{array}\right]X_2\\ q_2{X}_2+{X_2\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ B_f\left(\hat{\gamma }\left(t\right)\right)M_i\end{array}\right]}^T& -r_2{X}_2\end{array}\right]<0& \left(25\right)\end{array}$ $\begin{array}{cc}\left(\begin{array}{c}\sin {\theta }_2\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2+\\ {X_2\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T\end{array}\right\}\\ \cos {\theta }_2\left\{\begin{array}{c}{X_2\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T-\\ \left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2\end{array}\right\}\end{array}\begin{array}{c}\cos {\theta }_2\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2-\\ {X_2\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T\end{array}\right\}\\ \sin {\theta }_2\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2+\\ {X_2\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T\end{array}\right\}\end{array}\right)<0& \left(26\right)\end{array}$ selecting M.sub.j of a vertex corresponding to θ.sub.j as a parameter matrix of the virtual actuator; step 1.3.6: representing the parameter matrix N of the virtual actuator as:
N=B.sub.f.sup.†B.sub.p  (27) where † represents pseudo-inversion of the matrix; step 1.4: designing an interval error observer according to the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the system with fault; step 1.4.1: representing the reference model of the aircraft engine LPV model having disturbance and actuator and sensor faults as:
{dot over (x)}.sub.ref(t)=A.sub.0x.sub.ref(t)+B.sub.f({circumflex over (γ)}(t))u.sub.ref(t)
y.sub.ref(t)=C.sub.f({circumflex over (ϕ)}(t))x.sub.ref(t)  (28) where x.sub.ref ∈ R.sup.n.sup.x is a reference state vector of the aircraft engine LPV model having disturbance and actuator and sensor faults; u.sub.ref ∈ R.sup.n.sup.x is control input of the aircraft engine LPV model having disturbance and actuator and sensor faults; y.sub.ref ∈ R.sup.n.sup.y is a reference output vector of the aircraft engine LPV model having disturbance and actuator and sensor faults; step 1.4.2: defining an error e(t)=x.sub.ref(t)−x.sub.f(t) between the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the aircraft engine to obtain error state equations of the system with fault of the aircraft engine based on the LPV model:
ė(t)=[A.sub.0+ΔA(θ)]e(t)+B.sub.f({circumflex over (γ)})Δu(t)−B.sub.f(Δγ)u.sub.f(t)−ΔA(θ)x.sub.ref(t)−d.sub.f(t)
ε.sub.c(t)=C.sub.f(ϕ(t))e(t)−C.sub.f(Δϕ)x.sub.ref(t)−v(t)  (29) where Δu(t) and ε.sub.c(t) represent the input and output difference between the reference model and faulty aircraft engine LPV model with Δu(t)=u.sub.ref(t)−u.sub.f(t) and ε.sub.c(t)=y.sub.ref(t)−y.sub.f(t); step 1.4.3: representing state equations of an upper bound ē and a lower bound ē of the error e between the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the aircraft engine as:
{dot over (ē)}(t)=[A.sub.0−LC.sub.f(ϕ(t))]ē(t)+[B.sub.0+ΔB]Δu.sub.c(t)+L[ε.sub.c(t)+C.sub.px.sub.va+(C.sub.p−PC.sub.f(ϕ(t)))x.sub.vs]+|L|V−d(t)+ΔA|x.sub.ref(t)|+ΔB|u.sub.ref|+ϕ(t)
{dot over (e)}(t)=[A.sub.0−LC.sub.f(ϕ(t))]e(t)+[B.sub.0−ΔB]Δu.sub.c(t)+L[ε.sub.c(t)+C.sub.px.sub.va+(C.sub.p−PC.sub.f(ϕ(t)))x.sub.vs]−|L|V−d(t)−ΔA|x.sub.ref(t)|−ΔB|u.sub.ref|−ϕ(t)  (30) where ϕ(t)=ΔA(ē.sub.v.sup.+(t)+e.sub.v.sup.−(t)), e.sub.v is a difference among the error state variable of the system with fault of the aircraft engine based on the LPV model, the state variable of the virtual actuator and the state variable of the virtual sensor; the upper bound of e.sub.v is ē.sub.v(t)=ē(t)−x.sub.va(t)−x.sub.vs(t); the lower bound of e.sub.v is e.sub.v(t)=e(t)−x.sub.va(t)−x.sub.vs(t); A.sub.0−LC.sub.f ∈ M.sup.n.sup.x.sup.×n.sup.x; step 1.4.4: setting e.sub.a=0.5(ē+e),e.sub.d=ē−e, and obtaining the interval error observer from (30);
ė.sub.d(t)=[A.sub.0−LC.sub.f(ϕ(t))]e.sub.d(t)+2ΔBΔu.sub.c(t)+ϕ.sub.d(t)+δ.sub.d(t)
ė.sub.a(t)=[A.sub.0−LC.sub.f]e.sub.a(t)+B.sub.0K.sub.aE.sub.a(t)+B.sub.0K.sub.dE.sub.d(t)+δ.sub.a(t)+LC.sub.px.sub.va+L(C.sub.p−PC.sub.f)+LC.sub.fe(t)  (31) where ϕ.sub.d, δ.sub.d (t) and δ.sub.a(t) represent equivalent range of e.sub.v, range of the interval of external disturbance v(t) and d(t), and middle value of the interval of external disturbance v(t) and d(t), respectively;
ϕ.sub.d(t)=2ΔA(ē.sub.v.sup.+(t)+e.sub.v.sup.−(t))
δ.sub.d(t)=2|L|V−d(t)+d(t)+2ΔA|x.sub.ref(t)|2ΔB|u.sub.ref(t)|
δ.sub.a(t)=−Lv(t)−0.5(d(t)+d(t))  (32) step 1.5: using the aircraft engine state variable x.sub.f(t) of the aircraft engine LPV model having disturbance and actuator and sensor faults, the output variable y.sub.f(t), the reference model state variable x.sub.ref(t) of the system with fault, the virtual actuator state variable x.sub.va(t) and the virtual sensor state variable x.sub.vs(t) as inputs of the interval error observer; using the interval error observer output e.sub.a(t), e.sub.d(t) as the input of the error feedback controller; using the error feedback controller output Δu.sub.c(t) as the input of the virtual actuator; inputting the difference between the reference model output u.sub.ref(t) of the system with fault and the virtual actuator output Δu(t) as a control signal into the controller having the system with fault of the aircraft engine, thereby realizing active fault tolerant control of the aircraft engine.

Description

DESCRIPTION OF DRAWINGS

(1) FIG. 1 is an overall structural diagram of a system.

(2) FIG. 2(a) and FIG. 2(b) are respectively contrasts of trajectories of H=0, Ma=0, n.sub.2=94% aircraft engine LPV model states x.sub.p1(t) and x.sub.p2(t) and trajectories of fault-free reference model states x.sub.pref,1(t) and x.sub.pref,2(t).

(3) FIG. 3 is a flow chart of an error feedback controller algorithm.

(4) FIG. 4(a) and FIG. 4(b) are respectively the estimated curves of error states e.sub.p1(t) and e.sub.p2(t), upper bound states ē.sub.p1(t) and ē.sub.p2(t) and lower bound states e.sub.p1(t) and e.sub.p2(t) of H=0, Ma=0, n.sub.2=94% aircraft engine fault-free system.

(5) FIG. 5 is a varying curve of an actuator fault factor γ.sub.i and a sensor fault factor ϕ.sub.1.

(6) FIG. 6(a) and FIG. 6(b) are respectively the contrasts of trajectories of aircraft engine states x.sub.f1(t) and x.sub.f2(t) at H=0, Ma=0, n.sub.2=94% under both disturbances and actuator and sensor faults, and trajectories of fault-free reference model states x.sub.pref,1(t) and x.sub.pref,2(t).

(7) FIG. 7(a) and FIG. 7(b) are respectively the estimated curves of aircraft engine error states e.sub.pf1(t) and e.sub.pf2(t), upper bound states ē.sub.p1(t) and ē.sub.p2(t) and lower bound states e.sub.p1(t) and e.sub.p2(t) at H=0, Ma=0, n.sub.2=94% under both disturbances and actuator and sensor faults.

(8) FIG. 8 is a flow chart of a virtual sensor algorithm.

(9) FIG. 9 is a flow chart of a virtual actuator algorithm.

(10) FIG. 10 is a flow chart of an interval error observer algorithm.

(11) FIG. 11(a) and FIG. 11(b) are respectively the contrasts of trajectories of aircraft engine states x.sub.1(t) and x.sub.2(t) at H=0, Ma=0, n.sub.2=94% after active fault tolerant control and trajectories of fault reference model states x.sub.ref,1(t) and x.sub.ref,2(t).

(12) FIG. 12(a) and FIG. 12(b) are respectively the estimated curves of aircraft engine error states e.sub.1(t) and e.sub.2(t), and upper bound states ē.sub.1(t) and ē.sub.2 (t) and lower bound states e.sub.1(t) and e.sub.2(t) of an error observer at H=0, Ma=0, n.sub.2=94% after active fault tolerant control.

DETAILED DESCRIPTION

(13) The embodiments of the present invention will be further described in detail below in combination with the drawings and the technical solution.

(14) The overall structure of the present invention is shown in FIG. 1, and comprises the following specific steps:

(15) step 1.1: establishing an affine parameter-dependent aircraft engine LPV model; and taking relative conversion speed n.sub.2 of a high pressure rotor of the aircraft engine as a variable parameter θ to normalizing the speed n.sub.2=88%, 89%, . . . ,100%, i.e., θ ∈[−1,1], to obtain a model:

(16) $$\begin{gathered} \begin{array}{cc}{\stackrel{.}{x}}_p=\left[A_0+\Delta A\left(\theta \right)\right]x_p\left(t\right)+\left[B_0+\Delta B\left(\theta \right)\right]u_p\left(t\right)+d_f\left(t\right) \\ {y}_p=C_p{x}_p\left(t\right)+v\left(t\right)\mathrm{where}& \left(33\right)\end{array} \end{gathered}$$$$\begin{gathered} \begin{array}{cc}{A}_0=\left[\begin{array}{cc}-2.6748& -0.6877\\ 1.0704& -4.4672\end{array}\right],\Delta A\left(\theta \right)=\left[\begin{array}{cc}0.5199\theta & -2.4061\theta \\ 0.1049\theta & -0.8365\theta \end{array}\right] \\ {B}_0=\left[\begin{array}{c}0.0033\\ 0.0012\end{array}\right],\Delta B\left(\theta \right)=\left[\begin{array}{c}-0.0004\theta \\ -0.0001\theta \end{array}\right] \\ {C}_p=\left[\begin{array}{cc}0& 1\end{array}\right]& \left(34\right)\end{array} \end{gathered}$$
the state variable initial value is x.sub.p(0)=[0, 0].sup.T; the upper bound and the lower bound of a disturbance variable d.sub.f(t) are d, d ∈ R.sup.n.sup.x;

(17) $\overline{d}=-\underset{¯}{d}=\left[\begin{array}{c}0.001\\ 0.001\end{array}\right];$
and the sensor noise bound is V=0.01. ΔA(θ) and ΔB(θ) have established −ΔA≤ΔA(θ)≤ΔA and −ΔB≤ΔB(θ)≤ΔB.

(18) $\begin{array}{cc}\stackrel{\_}{\Delta A}=\left[\begin{array}{cc}0.5199& 2.4061\\ 0.1049& 0.8365\end{array}\right],\stackrel{\_}{\Delta B}=\left[\begin{array}{c}0.0004\\ 0.0001\end{array}\right]& \left(35\right)\end{array}$

(19) Step 1.2: representing the reference model of the fault-free system of the aircraft engine as

(20) $\begin{array}{cc}L=\left[\begin{array}{c}-5\\ 20\end{array}\right]& \left(38\right)\end{array}$

(21) where the state vector of the reference model is a constant value x.sub.pref(t)=[4, 2].sup.T. At H=0, Ma=0 and n.sub.2=94%, contrasts of trajectories of aircraft engine LPV model states x.sub.p1(t) and x.sub.p2(t) and trajectories of fault-free reference model states x.sub.pref,1(t) and x.sub.pref,2(t) are shown in FIG. 2. An error feedback controller of a fault-free system of the aircraft engine is designed, and an algorithm flow of the error feedback controller is shown in FIG. 3.

(22) Step 1.2.1: defining an error e.sub.p(t)=x.sub.pref−x.sub.pref,2 between the affine parameter-dependent aircraft engine LPV model and the reference model of the fault-free system of the aircraft engine, with an initial value is e.sub.p(0)=x.sub.pref(0)−x.sub.p(0)=[4, 2].sup.T.

(23) Step 1.2.2: representing state equations of the upper bound e.sub.P and the lower bound of the error vector e.sub.p as:
{dot over (ē)}.sub.p(t)=[A.sub.0−LC.sub.p]ē.sub.p(t)+[B.sub.0+ΔB]Δu.sub.cp(t)+Lε.sub.cp(t)+|L|V−d(t)+ΔA|x.sub.pref(t)|+ϕ.sub.p(t)
{dot over (e)}.sub.p(t)=[A.sub.0−LC.sub.p]e.sub.p(t)+[B.sub.0−ΔB]Δu.sub.cp(t)+Lε.sub.cp(t)−|L|V−d(t)−ΔA|x.sub.pref(t)|−ϕ.sub.p(t)  (37)

(24) where e.sub.p(0)=[−50, −50].sup.T, e.sub.p(0)=[50,50].sup.T and ϕ.sub.p(t)=ΔA(ē.sub.p.sup.+(t)+e.sub.p.sup.−(t)). At H=0, Ma=0 and n.sub.2=94%, the estimated curves of error states e.sub.p1(t) and e.sub.p2(t), upper bound ē.sub.p1(t) and ē.sub.p2(t) and lower bound e.sub.p1(t) and e.sub.p2(t) of aircraft engine fault-free system are shown in FIG. 4. From A.sub.0−LC.sub.p ∈ M.sup.n.sup.x.sup.×n.sup.r, the error gain matrix of the fault-free system can be obtained

(25) Step 1.2.3: respectively setting e.sub.pa=0.5(ē.sub.p+e.sub.p), e.sub.pd=ē.sub.p−e.sub.p to obtain
ė.sub.pd(t)=[A.sub.0−LC.sub.p]e.sub.pd(t)+2ΔBΔu.sub.cp(t)+ϕ.sub.pd(t)+δ.sub.pd(t)
ė.sub.pa(t)=[A.sub.0−LC.sub.p]e.sub.pa(t)+B.sub.0K.sub.ae.sub.pa(t)+B.sub.0K.sub.de.sub.pd(t)+LC.sub.pe.sub.p(t)+δ.sub.pa(t)
ϕ.sub.pd(t)=2ΔA(ē.sub.p.sup.+(t)+e.sub.p.sup.−(t))
δ.sub.pd(t)=2|L|V−d(t)+d(t)+2ΔA|x.sub.pref(t)|
δ.sub.pa(t)=−Lv(t)−0.5(d(t)+d(t))  (39)
where ē.sub.p1 and e.sub.p1 (ē.sub.p2 and e.sub.p2) respectively represent the first (second) element of ē.sub.p and e.sub.p, and e.sub.x,1 and e.sub.x,2 respectively represent the first element and the second element of e.sub.x.

(26) 0$$\begin{gathered} \begin{array}{cc}2\left({\overline{e}}_{p1}^{+}+{\underset{\_}{e}}_{p1}^{-}\right)=\{\begin{array}{cc}2{\overline{e}}_{p1}=e_{\mathrm{pd},1}+2e_{\mathrm{pa},1}& {\overline{e}}_{p1}\ge 0,{\underset{\_}{e}}_{p1}\ge 0\\ 2\left({\overline{e}}_{p1}-{\underset{\_}{e}}_{p1}\right)=2e_{pd,1}& {\overline{e}}_{p1}\ge 0,{\underset{\_}{e}}_{p1}<0\\ -2{\underset{\_}{e}}_{p1}=e_{\mathrm{pd},1}-2e_{\mathrm{pa},1}& {\overline{e}}_{p1}<0,{\underset{\_}{e}}_{p1}<0\end{array} \\ 2\left({\overline{e}}_{p2}^{+}+{\underset{\_}{e}}_{p2}^{-}\right)=\{\begin{array}{cc}2{\overline{e}}_{\mathrm{p2}}=e_{\mathrm{pd},2}+2e_{\mathrm{pa},2}& {\overline{e}}_{p2}\ge 0,{\underset{\_}{e}}_{p2}\ge 0\\ 2\left({\overline{e}}_{p2}-{\underset{\_}{e}}_{p2}\right)=2e_{\mathrm{pd},2}& {\overline{e}}_{p2}\ge 0,{\underset{\_}{e}}_{p2}<0\\ -2{\underset{\_}{e}}_{p1}=e_{\mathrm{pd},2}-2e_{\mathrm{pa},2}& {\overline{e}}_{p2}<0,{\underset{\_}{e}}_{p2}<0\end{array}& \left(40\right)\end{array} \end{gathered}$$

(27) Step 1.2.4: representing the output of the error feedback controller as
Δu.sub.cp(t)=K.sub.ae.sub.pa(t)+K.sub.de.sub.pd(t)  (41)

(28) representing the gain matrix of the error feedback controller as K.sub.d, K.sub.a ∈ R.sup.n.sup.x.sup.×n.sup.x; setting

(29) $$\begin{gathered} \begin{array}{cc}{e}_x\left(t\right)=e_p\left(t\right)-e_{\mathrm{pa}}\left(t\right),-0.5e_{pd}\left(t\right)\le e_x\left(t\right)\le 0.5e_{pd}\left(t\right),\mathrm{and}\mathrm{then} \\ {\stackrel{.}{e}}_{\mathrm{pa}}\left(t\right)=\left[A_0+B_0{K}_a\right]e_{\mathrm{pa}}\left(t\right)+B_0{K}_d{e}_{pd}\left(t\right)+L{C}_p{e}_x\left(t\right)+{\delta }_{\mathrm{pa}}\left(t\right)& \left(42\right)\end{array} \end{gathered}$$

(30) Step 1.2.5: rewriting (39) and (42) as

(31) ${\dot{\xi }}_p\left(t\right)=G_p\left(t\right){\xi }_p\left(t\right)+{\delta }_p\left(t\right)$$G_p\left(t\right)=\left[\begin{array}{cc}{A}_0-L{C}_p& 0\\ B_0{K}_d& A_0+B_0{K}_a\end{array}\right]+A_{pd}\left(t\right)$

(32) where

(33) ${\xi }_p\left(t\right)={\left[{e_{pd}\left(t\right)}^T,{e_{\mathrm{pa}}\left(t\right)}^T\right]}^T{\delta }_p\left(t\right)={\left[{\left({\delta }_{pd}\left(t\right)+2\stackrel{\_}{\Delta B}\Delta u_{cp}\left(t\right)\right)}^T,{{\delta }_{\mathrm{pa}}\left(t\right)}^T\right]}^T,$
and then

(34) $\begin{array}{cc}\begin{array}{c}\left[\begin{array}{c}{\varphi }_{pd}\\ L{C}_p{e}_x\end{array}\right]=A_{pd}\left[\begin{array}{c}{e}_{pd}\\ e_{\mathrm{pa}}\end{array}\right]\\ 2\stackrel{\_}{\Delta A}\left({\overline{e}}_p^{+}\left(t\right)+{\underset{\_}{e}}_p^{-}\left(t\right)\right)=A_{pd1}\left[\begin{array}{c}{e}_{pd}\\ e_{\mathrm{pa}}\end{array}\right],{\mathrm{LC}}_p{e}_x=A_{pd2}\left[\begin{array}{c}{e}_{pd}\\ e_{\mathrm{pa}}\end{array}\right]\\ A_{pd1}=\stackrel{\_}{\Delta A}\left[\begin{array}{cccc}{a}_{11}& 0& a_{13}& 0\\ 0& a_{22}& 0& a_{24}\end{array}\right],A_{pd2}=\left[\begin{array}{cccc}0& a_{31}& 0& 0\\ 0& a_{41}& 0& 0\end{array}\right]\\ A_{pd}=\left[\begin{array}{c}{A}_{pd1}\\ A_{pd2}\end{array}\right]\end{array}& \left(45\right)\end{array}$

(35) All possible combining forms are considered: (a.sub.11,a.sub.13) ∈ {(1,2),(2,0),(1,−2)}, (a.sub.22,a.sub.24) ∈ {(1,2),(2,0),(1,−2)} and (a.sub.31,a.sub.41) ∈ {(−2.5,10),(2.5,−10)}.

(36) Step 1.2.6: S.sup.m×m representing an m-dimensional real symmetric square matrix; setting a matrix E,F ∈ S.sup.2n.sup.x.sup.×2n.sup.x; E,F 0 representing that each element in E,F is greater than 0; constant λ>0; and obtaining a matrix inequality:
G.sub.p.sup.TE+EG.sub.p+λE+F0  (46)
namely, setting each element in G.sub.p.sup.TE+EG.sub.p+λE+F to be less than 0; converting the matrix inequality (46) to a linear matrix inequality (LMI), and multiplying the left and right sides of the inequality (46) by to obtain

(37) $\begin{array}{cc}{E}^1{G}_p^T+G_p{E}^1+\lambda E^1+F_p\prec 0& \left(47\right)\end{array}$$$\begin{gathered} \begin{array}{cc}{G}_p\left(t\right)=\left[\begin{array}{cc}{A}_0-L{C}_p& 0\\ 0& A_0\end{array}\right]+A_{pd}\left(t\right)+\left[\begin{array}{c}0\\ B_0\end{array}\right]K \\ K=\left[\begin{array}{cc}{K}_d& K_a\end{array}\right]& \left(48\right)\end{array} \end{gathered}$$
introducing W=KE.sup.−1, and then converting inequality (46) into the LMI; using an LMI tool kit to obtain
K.sub.d=[−0 0.0014 −0.0002]
K.sub.a=[−14.5130 −21.8837]

(38) Step 1.3: describing the aircraft engine LPV model having disturbance and actuator and sensor faults as:
{dot over (x)}.sub.f(t)=[A.sub.0+ΔA(θ)]x.sub.f(t)+B.sub.f(γ(t))u.sub.f(t)+d.sub.f(t)
y.sub.f(t)=C.sub.f(ϕ(t))x.sub.f(t)+v(t)
B.sub.f(γ(t))=[B.sub.0+ΔB(θ)]diag(γ.sub.1(t), . . . ,γ.sub.n(t))
C.sub.f(ϕ(t))=C.sub.p diag(ϕ.sub.1(t), . . . ,ϕ.sub.n(t))  (50)

(39) where state variable initial values x.sub.f (θ)=[0, 0].sup.T, B.sub.f(γ(t)) and C.sub.f(ϕ(t)) are respectively actuator and sensor faults; and the actuator fault factor γ.sub.1 and the sensor fault factor ϕ.sub.1 decay from 1 to 0.2 in the 5th to 6th seconds, as shown in FIG. 5. At H=0, Ma=0 and n.sub.2=94%, contrasts of trajectories of states x.sub.f1(t) and x.sub.f2(t) of the aircraft engine having disturbance and actuator and sensor faults and trajectories of fault-free reference model states x.sub.pref,1(t) and x.sub.pref,2 (t) are shown in FIG. 6. Estimated curves of error states e.sub.pf1(t) and e.sub.pf2(t) and upper bounds ē.sub.p1(t) and ē.sub.p2(t) and lower bounds e.sub.p1(t) and e.sub.p2(t) of the aircraft engine having disturbance and actuator and sensor faults are shown in FIG. 7. A virtual sensor and a virtual actuator are respectively designed according to the actuator and sensor faults, and algorithm flows are respectively shown in FIG. 8 and FIG. 9.

(40) Step 1.3.1: designing the virtual sensor as
{dot over (x)}.sub.vs(t)=A.sub.vs(θ)x.sub.vs(t)+B.sub.f({circumflex over (γ)}(t))Δu(t)+Qy.sub.f(t)
{circumflex over (γ)}.sub.f(t)=C.sub.vsx.sub.vs(t)+Py.sub.f(t)  (51)
where
A.sub.vs(θ)=A.sub.0+ΔA(θ)−QC.sub.f({circumflex over (ϕ)}(t))
C.sub.vs=C.sub.p−PC.sub.f({circumflex over (ϕ)}(t))  (52)

(41) where x.sub.vs ∈ R.sup.n.sup.x is a state variable of a virtual sensor system; Δu ∈ R.sup.n.sup.x is a difference in inputs of a fault model and a fault reference model; γ.sub.f ∈ R.sup.n.sup.y is an output vector of the virtual sensor system; Q and P are respectively parameter matrices of the virtual sensor.

(42) Step 1.3.2: selecting an LMI region S.sub.1(10,−4.5,15,π/6) and solving LMIs (53)-(55)

(43) $\begin{array}{cc}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1+{X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T+2{\rho }_1{X}_1<0& \left(53\right)\end{array}$$\begin{array}{cc}\left[\begin{array}{cc}-r_1{X}_1& q_1{X}_1+\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\end{array}\right]X_1\\ q_1{X}_1+{X_1\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\end{array}\right]}^T& -r_1{X}_1\end{array}\right]<0& \left(54\right)\end{array}$$\begin{array}{cc}\left(\begin{array}{c}\sin {\theta }_1\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1+\\ {X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T\end{array}\right\}\\ \cos {\theta }_1\left\{\begin{array}{c}{X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T-\\ \left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1\end{array}\right\}\end{array}\begin{array}{c}\cos {\theta }_1\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1-\\ {X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T\end{array}\right\}\\ \sin {\theta }_1\left\{\begin{array}{c}\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]X_1+\\ {X_1\left[A_0+\Delta A\left({\theta }_j\right)-Q_j{C}_f\left(\hat{\varphi }\left(t\right)\right)\right]}^T\end{array}\right\}\end{array}\right)<0& \left(55\right)\end{array}$

(44) obtaining a parameter matrix of a virtual sensor of a corresponding vertex
Q.sub.1=[−15.4224; 24.4935]
Q.sub.2=[8.5894; 33.1359]  (56)

(45) Step 1.3.3: representing the parameter matrix P of the virtual sensor as
P=C.sub.pC.sub.f.sup.†  (57)

(46) where † represents pseudo-inversion of the matrix;

(47) step 1.3.4: designing the virtual actuator as
{dot over (x)}.sub.va(t)=A.sub.vax.sub.va(t)+B.sub.vaΔu.sub.c(t)
Δu(t)=Mx.sub.va(t)+NΔu.sub.c(t)
y.sub.c(t)=ŷ.sub.f(t)+C.sub.px.sub.va(t)  (58)
where
A.sub.va=A.sub.0+ΔA(θ)−B.sub.f({circumflex over (γ)}(t))M
B.sub.va=B.sub.0+ΔB(θ)−B.sub.f({circumflex over (γ)}(t))N  (59)

(48) where x.sub.va ∈ R.sup.n.sup.x is a state variable of the virtual actuator system; Δu.sub.c ∈ R.sup.n.sup.x is the output of the error feedback controller; y.sub.c ∈ R.sup.n.sup.y is an output vector of the virtual actuator system; M and N are respectively parameter matrices of the virtual actuator;

(49) Step 1.3.5: selecting an LMI region S.sub.2(1.5,−2,8,π/6) and solving LMIs (60)-(62)

(50) $\begin{array}{cc}\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2+{X_2\left[A_0+\Delta A\left({\theta }_j\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T+2{\rho }_2{X}_2<0& \left(60\right)\end{array}$$\begin{array}{cc}\left[\begin{array}{cc}-r_2{X}_2& q_2{X}_2+\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ B_f\left(\hat{\gamma }\left(t\right)\right)M_i\end{array}\right]X_2\\ q_2{X}_2+{X_2\left[\begin{array}{c}{A}_0+\Delta A\left({\theta }_j\right)-\\ B_f\left(\hat{\gamma }\left(t\right)\right)M_i\end{array}\right]}^T& -r_2{X}_2\end{array}\right]<0& \left(61\right)\end{array}$$\begin{array}{cc}\left(\begin{array}{c}\sin {\theta }_2\left\{\begin{array}{c}\left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2+\\ {X_2\left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T\end{array}\right\}\\ \cos {\theta }_2\left\{\begin{array}{c}{X_2\left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T-\\ \left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2\end{array}\right\}\end{array}\begin{array}{c}\cos {\theta }_2\left\{\begin{array}{c}\left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2-\\ {X_2\left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T\end{array}\right\}\\ \sin {\theta }_2\left\{\begin{array}{c}\left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]X_2+\\ {X_2\left[A_0+\Delta A\left(\rho \left(t\right)\right)-B_f\left(\hat{\gamma }\left(t\right)\right)M_i\right]}^T\end{array}\right\}\end{array}\right)<0& \left(62\right)\end{array}$

(51) obtaining a parameter matrix of a virtual actuator of a corresponding vertex
M.sub.1=[3690.6 −4333.2]
M.sub.2=[2170.5 2186.6]  (63)

(52) Step 1.3.6: representing the matrix N of the virtual actuator as
N=B.sub.f.sup.†B.sub.P=5  (64)

(53) Step 1.4: designing an interval error observer, wherein an algorithm flow of the interval error observer is shown in FIG. 10.

(54) Step 1.4.1: representing the reference model of the aircraft engine system having disturbance and actuator and sensor faults as
{dot over (x)}.sub.ref(t)=A.sub.0x.sub.ref(t)+B.sub.f({circumflex over (γ)}(t))u.sub.ref(t)
y.sub.ref(t)=C.sub.f({circumflex over (ϕ)}(t))x.sub.ref(t)  (65)

(55) where the state vector of the reference model is a constant value x.sub.ref(t)=[4,2].sup.T.

(56) Step 1.4.2: defining an error e(t)=x.sub.ref−x.sub.f between the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the aircraft engine with an initial value of the error e(0)=x.sub.ref(0)−x.sub.f(0)=[4,2].sup.T. At H=0, Ma=0 and n.sub.2=94%, the contrasts of trajectories of aircraft engine states x.sub.1(t) and x.sub.2(t) and trajectories of fault reference model states x.sub.ref1(t) and x.sub.ref,2(t) after active fault tolerant control are shown in FIG. 11.

(57) Step 1.4.3: representing state equations of an upper bound ē and a lower bound e of the error e between the aircraft engine LPV model having disturbance and actuator and sensor faults and the reference model of the aircraft engine as
{dot over (ē)}(t)=[A.sub.0−LC.sub.f(ϕ(t))]ē(t)+[B.sub.0+ΔB]Δu.sub.c(t)+L[ε.sub.c(t)+C.sub.px.sub.va+(C.sub.p−PC.sub.f(ϕ(t)))x.sub.vs]+|L|V−d(t)+ΔA|x.sub.ref(t)|+ΔB|u.sub.ref|+ϕ(t)
{dot over (e)}(t)=[A.sub.0−LC.sub.f(ϕ(t))]e(t)+[B.sub.0−ΔB]Δu.sub.c(t)+L[ε.sub.c(t)+C.sub.px.sub.va+(C.sub.p−PC.sub.f(ϕ(t)))x.sub.vs]−|L|V−d(t)−ΔA|x.sub.ref(t)|−ΔB|u.sub.ref|−ϕ(t)  (66)

(58) where ϕ(t)=ΔA(ē.sub.v.sup.+(t)+e.sub.v.sup.−(t)), e.sub.v(t)=ē(t)−x.sub.va(t)−x.sub.vs(t), e.sub.v(0)=[−50,−50].sup.T and ē.sub.v(0)=[50,50].sup.T. e.sub.v(t)=e(t)−x.sub.va(t)−x.sub.vs(t). The gain matrix L of the observer satisfies A.sub.0−LC.sub.p ∈ M.sup.n.sup.x.sup.×n.sup.x.

(59) Step 1.4.4: setting e.sub.a=0.5(ē+e), e.sub.d=ē−e, and obtaining the interval error observer from (66)
ė.sub.d(t)=[A.sub.0−LC.sub.f(ϕ(t))]e.sub.d(t)+2ΔBΔu.sub.c(t)+ϕ.sub.d(t)+δ.sub.d(t)
ė.sub.a(t)=[A.sub.0−LC.sub.f]e.sub.a(t)+B.sub.0K.sub.aE.sub.a(t)+B.sub.0K.sub.dE.sub.d(t)+δ.sub.a(t)+LC.sub.px.sub.va+L(C.sub.p−PC.sub.f)+LC.sub.fe(t)  (67)
where
ϕ.sub.d(t)=2ΔA(ē.sub.v.sup.+(t)+e.sub.v.sup.−(t))
δ.sub.d(t)=2|L|V−d(t)+d(t)+2ΔA|x.sub.pref(t)|2ΔB|u.sub.pref(t)|
δ.sub.a(t)=−Lv(t)−0.5(d(t)+d(t))  (68)

(60) At H=0, Ma=0 and n.sub.2=94%, estimated curves of aircraft engine error states e.sub.1(t) and e.sub.2(t), and upper bound states ē.sub.1(t) and ē.sub.2(t) and lower bound states e.sub.1(t) and e.sub.2(t) of an error observer after active fault tolerant control are shown in FIG. 12.

(61) Step 1.5: showing the overall structure that realizes the active fault tolerant control of the aircraft engine in FIG. 1.

(62) Simulation results show that when the actuator and the sensor of the aircraft engine fail, an overshooting process occurs in states and outputs after active fault tolerant control, but the actuator and the sensor quickly return to a normal state. This indicates that the interval error observer-based aircraft engine active fault tolerant control method can ensure that the reconfigured system has the same performance criteria as the original fault-free system.