Bumpless transfer fault tolerant control method for aero-engine under actuator fault

Abstract

A bumpless transfer fault tolerant control method for aero-engine under actuator fault is disclosed. For an aero-engine actuator fault, by adopting an undesired oscillation problem produced by an active fault tolerant control method based on a virtual actuator, in order to solve the shortage of the existing control method, a bumpless transfer active fault tolerant control design method for the aero-engine actuator fault is provided, which can guarantee that a control system of the reconfigured aero-engine not only has the same state and output as an original fault-free system without changing the structure and parameters of a controller, to achieve a desired control objective, and that a reconfigured system has a smooth transient state, that is, output parameters such as rotational speed, temperature and pressure do not produce the undesired transient characteristics such as overshoot or oscillation.

Claims

1. A bumpless transfer fault tolerant control method for an aero-engine actuator fault, wherein comprising the following steps: step 1: expressing an aero-engine system as: $\begin{array}{cc}\hspace{1em}\{\begin{array}{c}\stackrel{.}{x}\left(t\right)=\mathrm{Ax}\left(t\right)+\mathrm{Bu}\left(t\right)\\ y\left(t\right)=\mathrm{Cx}\left(t\right)\end{array}& \left(1\right)\end{array}$ where, x(t)∈R.sup.n is a state of a system, A is n-dimensional square matrix, B is n×m matrix, C is n-dimensional square matrix, u(t)∈R.sup.m is a system input and the input is designed as a form of output-state feedback: m is control input dimension, and n is state dimension;
u(t)=−Ky(t)  (2) where, K is gain matrix of an aero-engine controller; when the actuator fault occurs, the aero-engine system is expressed as $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{x}}_f\left(t\right)={\mathrm{Ax}}_f\left(t\right)+B_f{u}_f\left(t\right)\\ y_f\left(t\right)={\mathrm{Cx}}_f\left(t\right)\end{array}& \left(3\right)\end{array}$ where, an actuator fault matrix B.sub.f is known, and B.sub.f.sup.T*B.sub.f is an invertible matrix; and f is used for characterizing a subscript of a fault system; step 2: designing an improved virtual actuator, with a structural form shown in (4): $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A\tilde{x}\left(t\right)+Bu\left(t\right)-B_f{u}_f\left(t\right)\\ u_f\left(t\right)=u_w\left(t\right)+N{u}_c\left(t\right)\\ y_c\left(t\right)=C\tilde{x}\left(t\right)+y_f\left(t\right)\end{array}& \left(4\right)\end{array}$ where, {tilde over (x)}(t)∈R.sup.n is a virtual actuator state, u.sub.c(t)=−Ky.sub.c(t),K is the same as that in an equation (2), u.sub.w(t) is a parameter to be designed, N=B.sub.f.sup.†B.sub.f, B.sub.f.sup.† is a Moore-Penrose inverse matrix of B.sub.f; c is a subscript of a nominal controller, and w is a subscript of a variable to be solved; step 3: in order to implement an aero-engine fault system in step 1 of a bumpless transfer of an improved virtual actuator in step 2, designing parameter u.sub.w(t) shown in an equation (5), wherein when the parameter u.sub.w (t) is optimized, the bumpless transfer of the virtual actuator in step 2 is implemented;
J=½{tilde over (x)}.sup.T(tf)C.sup.TRC{tilde over (x)}(tf)+∫.sub.0.sup.tf½(Bu(t)−B.sub.fu.sub.f(t)).sup.TP(Bu(t)−B.sub.fu.sub.f(t))+½{tilde over ({dot over (x)})}.sup.T(t)Q{tilde over ({dot over (x)})}dt  (5) where, J is the performance function, P≥0, Q≥0, R>0, P+Q>0, and P, Q, R are symmetric weight matrices; step 4: according to a form of an actuator fault matrix B.sub.f, considering the following two conditions:
Condition 1: B.sub.fB.sub.f.sup.†B=B  (6)
Condition 2: B.sub.fB.sub.f.sup.†B≠B  (7) when condition 1 occurs, the improved virtual actuator (4) in step 2 is simplified as a form of the following equation (8): $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A\tilde{x}\left(t\right)-B_f{u}_w\left(t\right)\\ \tilde{x}\left(t_0\right)=a\end{array}& \left(8\right)\end{array}$ where, a is an initial state that constant vectors characterize, which is obtained through difference between a state in aero-engine system (1) in step 1 and a state in a system (3) at the time when B.sub.f is diagnosed after the fault; when condition 2 occurs, the virtual actuator (4) in step 2 is written as a form of the following equation (9): $\begin{array}{cc}\hspace{1em}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A\tilde{x}\left(t\right)-\left(I-B_f{B}_f^{†}\right)\mathrm{BKC}\tilde{x}\left(t\right)-\left(I-B_f{B}_f^{†}\right){\mathrm{BKy}}_f\left(t\right)-B_f{u}_w\left(t\right)\\ \tilde{x}\left(t_0\right)=a\end{array}& \left(9\right)\end{array}$ where, a is an initial state that constant vectors characterize, which is obtained through difference between the state in aero-engine system (1) in step 1 and the state in a system (3) at the time when the fault B.sub.f is diagnosed, and I is n-dimensional square matrix; step 5: in consideration of the condition 1 in step 4, designing a parameter u.sub.w(t) according to an equation (10), that is, satisfying the parameter u.sub.w(t) in step 3 and implementing an aero-engine fault system (3) in step 1 of the bumpless transfer of the improved virtual actuator (4) in step 2:
u.sub.w(t)=(B.sub.f.sup.T(P+Q)B.sub.f).sup.−1B.sub.f.sup.T(QA+F(t)){tilde over (x)}(t)  (10) where, the matrix F (t) is a symmetric positive definite matrix, and satisfies the equation (11) in the time interval t∈[0, tf]:
−{dot over (F)}(t)=F(t)A+(A.sup.T−(A.sup.TQ+F(t))B.sub.f(B.sub.f.sup.T(P+Q)B.sub.f).sup.†B.sub.f.sup.T(QA+F(t)))   (11) F(t) satisfies the following boundary condition (12):
C.sup.TF(tf)C=R  (12) where, R is a weight matrix in step 3(5); step 6: in consideration of the condition 2 in step 4, defining {circumflex over (x)}(t):={tilde over (x)}(t)+x.sub.f(t), and expressing a reconfigured aero-engine control system as: $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\hat{x}}\left(t\right)=A\hat{x}\left(t\right)+Bu\left(t\right)\\ \hat{y}\left(t\right)=C\hat{x}\left(t\right)\end{array}& \left(13\right)\end{array}$ wherein, the initial state is {circumflex over (x)}(0)=x.sub.f(0)+{tilde over (x)}(0); and the reconfigured aero-engine control system state (14) influenced only by a design parameter K of an original aero-engine system controller is obtained by substituting an output-state feedback controller u (t)=−Kŷ(t)=−KC{circumflex over (x)}(t) into an equation (13), where K is consistent with the designed K in the equation (2) of step 1:
{circumflex over ({dot over (x)})}(t)=(A−BKC){circumflex over (x)}(t)  (14) the equation (14) is substituted into a virtual actuator structure (9) in step 4, to obtain: $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A\tilde{x}\left(t\right)-B_f{u}_w\left(t\right)-\left(I-B_f{B}_f^{†}\right)\mathrm{BKC}\hat{x}\left(t\right)\\ \tilde{x}\left(t_0\right)=a\end{array}& \left(15\right)\end{array}$ the parameter u.sub.w(t) is shown in an equation (16), that is, the performance index function in step 3 is satisfied, and the aero-engine fault system (3) in step 1 of the bumpless transfer of the improved virtual actuator (4) in step 2 is implemented:
u.sub.w(t)=(B.sub.f.sup.T(P+Q)B.sub.f).sup.−1B.sub.f.sup.T(−(P+Q)(I−B.sub.fB.sub.f.sup.†)BKC{circumflex over (x)}(t)+(QA+E(t)){tilde over (x)}(t)+G(t))  (16) where, {circumflex over (x)}(t) satisfies the equation (14), and E(t) is the symmetric positive definite matrix of the equation (17) and satisfies a boundary condition of the equation (18); $\begin{array}{cc}-\stackrel{.}{E}\left(t\right)=E\left(t\right)\left(I-{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^{T}Q\right)A+A^T\left(I-Q{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^T\right)E\left(t\right)-E\left(t\right){B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^{T}E\left(t\right)+A^T\mathrm{QA}-A^{T}Q{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^{T}QA& \left(17\right)\end{array}$ E(t) satisfies the boundary condition:
C.sup.TE(tf)C=R  (18) an adjoint vector G(t) satisfies the following equation: $\begin{array}{cc}\stackrel{.}{G}\left(t\right)=\left(\left(A^{T}Q+E\left(t\right)\right){B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}-A^T\right)G\left(t\right)+\left(E\left(t\right)+A^{T}Q\right)\left(I-{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}\left(P+Q\right)\right)\left(I-B_f{B}_f^{†}\right)\mathrm{KC}\hat{x}\left(t\right)& \left(19\right)\end{array}$ the boundary condition of the adjoint equation (19) is
G(tf)=0  (20); and step 7: controlling the aero-engine system using one of the improved virtual executor of equations (8) or (9) when the aero-engine actuator is faulty.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a flow chart of reconfiguration control design for a bumpless virtual actuator of an aero-engine under actuator fault;

(2) FIG. 2 is a virtual actuator switch framework of an aero-engine actuator fault system;

(3) FIG. 3 is a contrast diagram of bumpless transfer of reconfiguration control input [ΔW.sub.fb(t), ΔA.sub.8(t)].sup.T in a condition 1;

(4) FIG. 4 is a contrast diagram of bumpless transfer of reconfiguration control output [Δn.sub.l(t), Δn.sub.h(t)].sup.T in a condition 1;

(5) FIG. 5 is a contrast diagram of bumpless transfer of fuel flow W.sub.f of reconfiguration control input [Δn.sub.l(t), Δn.sub.h(t)].sup.T in a condition 2; and

(6) FIG. 6 is a contrast diagram of bumpless transfer of fuel flow W.sub.f of reconfiguration control output [Δn.sub.f(t), Δn.sub.c(t)].sup.T in a condition 2.

DETAILED DESCRIPTION

(7) The present invention will be further described below in combination with the drawings. The research object of the present invention is the reconfiguration and the switching process of a controller after an aero-engine actuator fault occurs, a design method thereof is shown in a flow chart of FIG. 1, and the detailed design steps are as follows:

(8) step 1: obtaining an aero-engine system model A,B,C,x(t.sub.0), a gain matrix K of an aero-engine controller and a parameter B.sub.f, x.sub.f(t.sub.0) of the aero-engine system after fault;

(9) step 2: according to an actuator parameter matrix B of the aero-engine system and the diagnosed actuator parameter matrix B.sub.f after fault, judging the conditions; if B.sub.fB.sub.f.sup.†B=B, performing a step 3; and if B.sub.fB.sub.f.sup.†B≠B, performing a step 5;

(10) step 3: designing a virtual actuator as:

(11) 0$\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A\tilde{x}\left(t\right)-B_f{u}_w\left(t\right)\\ u_f\left(t\right)=u_w\left(t\right)-B_f^{†}{B}_f{\mathrm{Ky}}_c\left(t\right)\\ y_c\left(t\right)=C\tilde{x}\left(t\right)+y_f\left(t\right)\end{array}& \left(21\right)\end{array}$

(12) where, {tilde over (x)}(t.sub.0)=x(t.sub.0)−x.sub.f(t.sub.0), u.sub.w(t)=(B.sub.f.sup.T(P+Q)B.sub.f).sup.−1B.sub.f.sup.T(QA+F(t)){tilde over (x)}(t); and a symmetric positive definite matrix F(t) is obtained by solving a Riccati equation (22) in which the boundary conditions satisfy C.sup.TF(tf)C=R.
−{dot over (F)}(t)=F(t)A+(A.sup.T−(A.sup.TQ+F(t))B.sub.f(B.sub.f.sup.T(P+Q)B.sub.f).sup.†B.sub.f.sup.T(QA+F(t)))   (22)

(13) Using the switch logic in FIG. 2, the reconfigured u.sub.f is switched into a fault model, and the compensated controller input y.sub.c(t) is switched into an original aero-engine controller without changing the parameter of the original aero-engine controller K.

(14) step 4: designing the virtual controller as:

(15) $\hspace{1em}\begin{array}{cc}\{\begin{array}{c}{u}_f\left(t\right)=u_w\left(t\right)-B_f^{†}{B}_{f}K{y}_c\left(t\right)\\ y_c\left(t\right)=C\tilde{x}\left(t\right)+y_f\left(t\right)\\ \stackrel{.}{\stackrel{~}{x}}\left(t\right)=A\tilde{x}\left(t\right)-B_f{u}_w\left(t\right)-\left(I-B_f{B}_f^{†}\right)\mathrm{BKC}\hat{x}\left(t\right)\\ \stackrel{.}{\hat{x}}\left(t\right)=\left(A-\mathrm{BKC}\right)\hat{x}\left(t\right)\\ \hat{x}\left(t_0\right)=x_f\left(t_0\right)+\tilde{x}\left(t_0\right)\end{array}& \left(21\right)\end{array}$

(16) where, u.sub.w(t) is:
u.sub.w(t)=(B.sub.f.sup.T(P+Q)B.sub.f).sup.−1B.sub.f.sup.T(−(P+Q)(I−B.sub.fB.sub.f.sup.†)BKC{circumflex over (x)}(t)+(QA+E(t)){tilde over (x)}(t)+G(t))  (22)

(17) The symmetric positive definite matrix E(t) in an equation (22) is obtained by solving the equation (24) in which the boundary conditions satisfy the Riccati equation (23); and an adjoint vector G(t) is obtained by solving the equation (25) in which the boundary conditions satisfy the equation (26).

(18) $\begin{array}{cc}-\stackrel{.}{E}\left(t\right)=E\left(t\right)\left(I-{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^{T}Q\right)A+A^T\left(I-Q{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^T\right)E\left(t\right)-E\left(t\right){B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^{T}E\left(t\right)+A^{T}QA-A^{T}Q{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}{B}_f^T\mathrm{QA}& \left(23\right)\\ C^{T}E\left(\mathrm{tf}\right)C=R& \left(24\right)\\ \stackrel{.}{G}\left(t\right)=\left(\left(A^{T}Q+E\left(t\right)\right){B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}-A^T\right)G\left(t\right)+\left(E\left(t\right)+A^{T}Q\right)\left(I-{B_f\left(B_f^T\left(P+Q\right)B_f\right)}^{-1}\left(P+Q\right)\right)\left(I-B_f{B}_f^{†}\right)\mathrm{KC}\hat{x}\left(t\right)& \left(25\right)\\ G\left(\mathrm{tf}\right)=0& \left(26\right)\end{array}$

(19) Using the switch logic in FIG. 2, the reconfigured u.sub.f is switched into an aero-engine fault system, and the compensated controller input y.sub.c(t) is switched into the original aero-engine controller without changing the parameter of the original aero-engine controller K.

(20) step 5: respectively verifying the design of bumpless transfer control under two conditions, wherein in a condition 1, a system model at a certain steady point of a test-run state of a three ducts variable cycle engine is adopted, and the model coefficient of the three ducts variable cycle engine is:

(21) $\begin{array}{cc}A=\left[\begin{array}{cc}-6.5865& 21.8290\\ -0.6504& 0.2127\end{array}\right],B=\left[\begin{array}{cc}0.0754& 0.2371\\ 0.2629& 0.1484\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],x\left(0\right)={\left[-20,35\right]}^T& \left(27\right)\end{array}$

(22) The control input is u=[ΔW.sub.fb(t),ΔA.sub.8 (t)].sup.T, where ΔW.sub.fb is the variation of aero-engine fuel flow, and ΔA.sub.8 is the variation [Δn.sub.l(t),Δn.sub.h(t)].sup.T of an aero-engine guide vane angle; and where Δn.sub.l is the variation of the rotational speed of an aero-engine low pressure rotor, and Δn.sub.h is the variation of the rotational speed of an aero-engine high pressure rotor.

(23) Suppose the actuator fault occurs at t=0.5 s, B.sub.f is diagnosed at t=3 s.

(24) $\begin{array}{cc}{B}_f=\left[\begin{array}{cc}0.6198& 0.4772\\ 0.3233& 0.1434\end{array}\right]& \left(28\right)\end{array}$

(25) Through the virtual actuator design of step 3, an input curve of an aero-engine system after fault is shown in FIG. 3, and a model output is shown in FIG. 4. Compared with the prior art, the input designed in step 3 can effectively reduce the bump brought by the switching and realize the recovery of a bumpless aero-engine system in FIG. 4.

(26) step 6: respectively verifying the design of the bumpless transfer control under two conditions, wherein in a condition 2, a small perturbation model in a turbofan engine mode “FC01” of 90K is adopted, and the aero-engine system is:

(27) $\begin{array}{cc}A=\left[\begin{array}{cc}-3.8557& 1.4467\\ 0.4690& -4.7081\end{array}\right],B=\left[\begin{array}{c}230.6739\\ 653.5547\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],x\left(0\right)={\left[-80,-103.5\right]}^T& \left(27\right)\end{array}$

(28) The control input is u=W.sub.f, and W.sub.f is turbofan engine fuel flow, y=[Δn.sub.f(t),Δn.sub.c(t)].sup.T,where Δn.sub.f is the variation of the rotational speed of the fan of a turbofan engine, and Δn.sub.c is the variation of the rotational speed of a compressor of the turbofan engine.

(29) Suppose the actuator fault occurs at t=0.4 s, B.sub.f is diagnosed at t=0.8 s.

(30) $\begin{array}{cc}{B}_f=\left[\begin{array}{c}161.4717\\ -522.8438\end{array}\right]& \left(28\right)\end{array}$

(31) Through the virtual actuator design of step 4, the input curve of a system after fault is shown in FIG. 5, and the model output is shown in FIG. 6. Compared with the prior art, the input designed in step 5 can effectively reduce the bump brought by the switching and realize the recovery of a bumpless aero-engine control system in FIG. 6.