US 11,635,734 B2 · 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.

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)}p(t)=[A0+ΔA(θ)]xp(t)+[B0+ΔB(θ)]up(t)+df(t)
yp(t)=Cpxp(t)+v(t) (1)
where Rm and Rm×n respectively represent a m-dimensional real number column vector and a m-row n-column real matrix;
state vectors xp=[Ynl Ynh]T ∈ Rnx, Ynl and Ynh respectively represent variation of relative conversion speed of low pressure and high pressure rotors;
nx represents the dimension of a state variable x;
ny represents the dimension of an output vector y;
nu represents the dimension of control input up;
control input up=Upf ∈ Rnx is a fuel pressure step signal;
output vectors yp=Ynh ∈ Rny, A0 ∈ Rnx×nx, B0 ∈ Rnx×nx and Cp ∈ Rny×nx are known system constant matrices;
df(t) is a disturbance variable;
the relative conversion speed nh of the high pressure rotor of the aircraft engine is a scheduling parameter θ ∈ Rp;
system variable matrices ΔA(θ) and ΔB(θ) satisfy −ΔA≤ΔA(θ)≤ΔA and −ΔB≤ΔB(θ)≤ΔB;
ΔA ∈ Rnx×nx is an upper bound of ΔA(θ);
ΔB ∈ Rnx×nu is an upper bound of ΔB(θ);
ΔA≥0, ΔB≥0;
a state variable initial value xp(0) satisfies x0≤xp(0)≤x0;
x0, x0 ∈ Rnx are respectively known upper bound and lower bound of the state variable initial value xp(0);
d,d ∈ Rnx are known upper bound and lower bound of an unknown disturbance df(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)}pref(t)=A0xpref(t)+B0upref(t)
ypref(t)=Cpxpref(t) (2)
where xpref ∈ Rnx is a reference state vector of the fault-free system;
upref ∈ Rnx is control input of the fault-free system;
ypref ∈ Rny 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 ep(t)=xpref(t)−xp(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:
ėp(t)=[A0+ΔA(θ)]ep(t)+[B0+ΔB(θ)]Δucp(t)−ΔA(θ)xpref(t)−ΔB(θ)upref(t)−df(t)
εcp(t)=Cpep(t)−v(t) (3)
where Δucp(t) and εcp(t) represent the input and output difference between the reference model and aircraft engine LPV model with Δucp(t)=upref(t)−up(t) and εcp(t)=ypref(t)−yp(t), respectively;
step 1.2.2:
representing state equations of the upper bound ēp and the lower bound ep of the error vector ep as:
{dot over (ē)}p(t)=[A0−LCp]ēp(t)+[B0+ΔB]Δucp(t)+Lεcp(t)+|L|V−d(t)+ΔA|xpref(t)|+ϕp(t)
{dot over (e)}p(t)=[A0−LCp]ep(t)+[B0−ΔB]Δucp(t)+Lεcp(t)−|L|V−d(t)−ΔA|xpref(t)|−ϕp(t) (4)
where ēp, ep ∈ Rnx are respectively the upper bound and the lower bound of the error vector ep, i.e., ep(t)≤ep(t)≤ēp(t);
ϕp(t)=ΔA(ēp+(t)+ep−(t)), ēp+=max {0, ēp}, ēp−=ēp+−ēp, eP+=max {0, ēp}, ep−=ep+−ep;
L ∈ Rnx×ny is an error gain matrix of the fault-free system and satisfies A0−LCp ∈ Mnx×nx;
Mnx represents a set of nx-dimensional Metzler matrix;
|L| represents taking absolute values of all elements of the matrix L;
step 1.2.3:
respectively setting epa=0.5(ēp+ep) and epd=ēp−ep, which represent the middle value and range of the interval of ep, respectively;
rewriting the formula (4) as:
ėpd(t)=[A0−LCp]epd(t)+2ΔBΔucp(t)+ϕpd(t)+δpd(t)
ėpa(t)=[A0−LCp]epa(t)+B0Δucp(t)+LCpep(t)+δpa(t) (5)
where ϕpd(t), δpa(t) and δpd(t) are variables defined as
ϕpd(t)=2ΔA(ēp−(t)+ep−(t))
δpd(t)=2|L|V−d(t)+d(t)+2ΔA|xpref(t)|
δpa(t)=−Lv(t)−0.5(d(t)+d(t)) (6)
step 1.2.4:
defining output signal of the error feedback controller as:
Δucp(t)=Kaepa(t)+Kdepd(t) (7)
where Kd, Ka ∈ Rnx×nx represent gain matrices of the error feedback controller signal (7);
setting ex(t)=ep(t)−epa(t), −0.5epd(t)≤ex(t)≤0.5epd(t), and then
ėpa(t)=[A0+B0Ka]epa(t)+B0Kdepd(t)+LCpex(t)+δpa(t) (8)
step 1.2.5:
rewriting formulas (5) and (8) as:
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where ξp(t) is an error vector composed of the range of the error interval epd and middle value of the error interval epa with
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step 1.2.6:
Sm×m representing an m-dimensional real symmetric square matrix;
setting a matrix E,F ∈ SR2nx×2nx;
E,F 0 representing that each element in E,F is greater than 0;
constant λ>0;
and obtaining a matrix inequality:
GpTE+EGp+λE+F0 (12)
namely, setting each element in GpTE+EGp+λE+F to be less than 0;
solving the matrix inequality (12) to obtain the gain matrices Kd, Ka 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)}f(t)=[A0+ΔA(θ)]xf(t)+Bf(γ(t))uf(t)+df(t)
yf(t)=Cf(ϕ(t))xf+v(t) (13)
where xf ∈ Rnx is a state vector of a system with fault;
uf ∈ Rnx is the control input of the system with fault;
yf ∈ Rny is an output vector of the system with fault;
Bf(γ(t)) and Cf(ϕ(t)) are respectively actuator and sensor faults, expressed as
Bf(γ(t))=[B0+ΔB(θ)]diag(γ1(t), . . . ,γn(t))
Cf(ϕ(t))=Cp diag(ϕ1(t), . . . ,ϕn(t)) (14)
where 0≤yi(t)≤1 and 0≤ϕj(t)≤1 respectively represent the failure degree of the i th actuator and the j th sensor;
γi=1 and γi=0 respectively represent health and complete failure of the i th actuator;
ϕj is similar;
diag(γ1, γ2, . . . , γn) represents a diagonal matrix with diagonal elements γ1, γ2, . . . , γn;
diag(ϕ1, ϕ2, . . . , ϕn) is similar;
setting γ(t) and ϕ(t) estimated values respectively as {circumflex over (γ)}(t) and {circumflex over (ϕ)}(t), and then
Bf(γ(t))=Bf({circumflex over (γ)}(t))+Bf(Δγ(t))
Cf(ϕ(t))=Cf(ϕ(t))+Cf(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)}vs(t)=Avs(θ)xvs(t)+Bf({circumflex over (γ)}(t))Δu(t)+Qyf(t)
{circumflex over (γ)}f(t)=Cvsxvs(t)+Pyf(t) (16)
where
Avs(θ)=A0+ΔA(θ)−QCf({circumflex over (ϕ)}(t))
Cvs=Cp−PCf({circumflex over (ϕ)}(t)) (17)
where xvs ∈ Rnx is a state variable of a virtual sensor;
Δu ∈ Rnx is a difference in control inputs of a fault model and a fault reference model;
{circumflex over (γ)}f ∈ Rny 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 S1(ρ1, q1, r1, θ1) representing an intersection of a left half complex plane region with a bound of −ρ1, a circular region with a radius of r1 and a circle center of q1 and a fan region having an intersection angle θ1 with a negative real axis;
representing a state matrix Avs of the virtual sensor as a polytope structure;
Avsj=A0+ΔA(θj)−QjCf({circumflex over (ϕ)}(t)), where θj represents the value of the j th vertex θ;
Avsj represents the value of the state matrix Avs of the virtual sensor of the j th vertex;
a necessary and sufficient condition for eigenvalues of Avsj to be in S1(ρ1, q1, r1, θ1) is that there exists a symmetrical matrix X1>0 so that the linear matrix inequalities (18)-(20) are established, thereby obtaining a parameter matrix Qj of the virtual sensor of the corresponding vertex;
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selecting Qj of a vertex corresponding to θj as a parameter matrix of the virtual sensor;
step 1.3.3:
representing the parameter matrix P of the virtual sensor as:
P=CpCf† (21)
where † represents pseudo-inversion of the matrix;
step 1.3.4:
designing the virtual actuator as
{dot over (x)}va(t)=Avaxva(t)+BvaΔuc(t)
Δu(t)=Mxva(t)+NΔuc(t)
yc(t)=ŷf(t)+Cpxva(t) (22)
where
Ava=A0+ΔA(θ)−Bf({circumflex over (γ)}(t))M
Bva=B0+ΔB(θ)−Bf({circumflex over (γ)}(t))N (23)
where xva ∈ Rnx is a state variable of the virtual actuator;
Δuc ∈ Rnx is the output of the error feedback controller;
yc ∈ Rny 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 S2(ρ2, q2, r2, θ2) representing an intersection of a left half complex plane region with a bound of −ρ2, a circular region with a radius of r2 and a circle center of q2 and a fan region having an intersection angle θ2 with a negative real axis;
representing a state matrix Ava of the virtual actuator as a polytope structure;
Avaj=A0+ΔA(θj)−Bf({circumflex over (γ)}(t))Mj, where θj represents the value of the j th vertex θ;
Avaj represents the value of the state matrix Ava of the virtual actuator of the j th vertex;
a necessary and sufficient condition for eigenvalues of Avaj to be in S2(ρ2, q2, r2, θ2) is that there exists a symmetrical matrix X2>0 so that the linear matrix inequalities (24)-(26) are established, thereby obtaining a parameter matrix Mi of the virtual actuator;
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selecting Mj of a vertex corresponding to θj as a parameter matrix of the virtual actuator;
step 1.3.6:
representing the parameter matrix N of the virtual actuator as:
N=Bf†Bp (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)}ref(t)=A0xref(t)+Bf({circumflex over (γ)}(t))uref(t)
yref(t)=Cf({circumflex over (ϕ)}(t))xref(t) (28)
where xref ∈ Rnx is a reference state vector of the aircraft engine LPV model having disturbance and actuator and sensor faults;
uref ∈ Rnx is control input of the aircraft engine LPV model having disturbance and actuator and sensor faults;
yref ∈ Rny 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)=xref(t)−xf(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)=[A0+ΔA(θ)]e(t)+Bf({circumflex over (γ)})Δu(t)−Bf(Δγ)uf(t)−ΔA(θ)xref(t)−df(t)
εc(t)=Cf(ϕ(t))e(t)−Cf(Δϕ)xref(t)−v(t) (29)
where Δu(t) and εc(t) represent the input and output difference between the reference model and faulty aircraft engine LPV model with Δu(t)=uref(t)−uf(t) and εc(t)=yref(t)−yf(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)=[A0−LCf(ϕ(t))]ē(t)+[B0+ΔB]Δuc(t)+L[εc(t)+Cpxva+(Cp−PCf(ϕ(t)))xvs]+|L|V−d(t)+ΔA|xref(t)|+ΔB|uref|+ϕ(t)
{dot over (e)}(t)=[A0−LCf(ϕ(t))]e(t)+[B0−ΔB]Δuc(t)+L[εc(t)+Cpxva+(Cp−PCf(ϕ(t)))xvs]−|L|V−d(t)−ΔA|xref(t)|−ΔB|uref|−ϕ(t) (30)
where ϕ(t)=ΔA(ēv+(t)+ev−(t)), ev 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 ev is ēv(t)=ē(t)−xva(t)−xvs(t);
the lower bound of ev is ev(t)=e(t)−xva(t)−xvs(t);
A0−LCf ∈ Mnx×nx;
step 1.4.4:
setting ea=0.5(ē+e),ed=ē−e, and obtaining the interval error observer from (30);
ėd(t)=[A0−LCf(ϕ(t))]ed(t)+2ΔBΔuc(t)+ϕd(t)+δd(t)
ėa(t)=[A0−LCf]ea(t)+B0KaEa(t)+B0KdEd(t)+δa(t)+LCpxva+L(Cp−PCf)+LCfe(t) (31)
where ϕd, δd (t) and δa(t) represent equivalent range of ev, 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;
ϕd(t)=2ΔA(ēv+(t)+ev−(t))
δd(t)=2|L|V−d(t)+d(t)+2ΔA|xref(t)|2ΔB|uref(t)|
δa(t)=−Lv(t)−0.5(d(t)+d(t)) (32)
step 1.5:
using the aircraft engine state variable xf(t) of the aircraft engine LPV model having disturbance and actuator and sensor faults, the output variable yf(t), the reference model state variable xref(t) of the system with fault, the virtual actuator state variable xva(t) and the virtual sensor state variable xvs(t) as inputs of the interval error observer;
using the interval error observer output ea(t), ed(t) as the input of the error feedback controller;
using the error feedback controller output Δuc(t) as the input of the virtual actuator;
inputting the difference between the reference model output uref(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.

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

Life of this patent

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

Life of this patent

Abstract

Claims1