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, an aero-engine system is expressed as $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{x}}_f\left(t\right)={\mathrm{Ax}}_f\left(t\right)+B_f.Math.u_f\left(t\right)\\ y_f\left(t\right)={\mathrm{Cx}}_f\left(t\right)\end{array}& \left(3\right)\end{array}$ 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): $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A.Math.\tilde{x}\left(t\right)+B.Math.u\left(t\right)-B_f.Math.u_f\left(t\right)\\ u_f\left(t\right)=u_w\left(t\right)+N.Math.u_c\left(t\right)\\ y_c\left(t\right)=C.Math.\tilde{x}\left(t\right)+y_f\left(t\right)\end{array}& \left(4\right)\end{array}$ 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 performance parameters shown in an equation (5), wherein when a performance function 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, 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 executor (4) in step 2 is simplified as a form of the following equation (8): $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A.Math.\tilde{x}\left(t\right)-B_f.Math.u_w\left(t\right)\\ \tilde{x}\left(t_0\right)=a\end{array}& \left(8\right)\end{array}$ 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 executor (4) in step 2 is written as a form of the following equation (9): $\begin{array}{cc}\hspace{1em}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A.Math.\tilde{x}\left(t\right)-\left(I-B_f.Math.B_f^{†}\right).Math.\mathrm{BKC}.Math.\tilde{x}\left(t\right)-\left(I-B_f.Math.B_f^{†}\right).Math.{\mathrm{BKy}}_f\left(t\right)-B_f.Math.u_w\left(t\right)\\ \tilde{x}\left(t_0\right)=a\end{array}& \left(9\right)\end{array}$ 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 a performance index function 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 the reconfigured aero-engine control system as: $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\hat{x}}\left(t\right)=A.Math.\hat{x}\left(t\right)+B.Math.u\left(t\right)\\ \hat{y}\left(t\right)=C.Math.\hat{x}\left(t\right)\end{array}& \left(13\right)\end{array}$ 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:
{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: $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A.Math.\tilde{x}\left(t\right)-B_f.Math.u_w\left(t\right)-\left(I-B_f.Math.B_f^{†}\right).Math.\mathrm{BKC}.Math.\hat{x}\left(t\right)\\ \tilde{x}\left(t_0\right)=a\end{array}& \left(15\right)\end{array}$ the design 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); $\begin{array}{cc}-\stackrel{.}{E}\left(t\right)=E\left(t\right).Math.\left(I-{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T.Math.Q\right).Math.A+A^T\left(I-Q.Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T\right).Math.E\left(t\right)-E\left(t\right).Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T.Math.E\left(t\right)+A^T.Math.\mathrm{QA}-A^T.Math.Q.Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T.Math.Q.Math.A& \left(17\right)\end{array}$ E (t) satisfies the boundary condition:
C.sup.TE(tf)C=R  (18) an adjoint vector G(t) satisfies the following equation: $\begin{array}{cc}\stackrel{.}{G}\left(t\right)=\left(\left(A^T.Math.Q+E\left(t\right)\right).Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}-A^T\right).Math.G\left(t\right)+\left(E\left(t\right)+A^T.Math.Q\right).Math..Math..Math.\left(I-{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.\left(P+Q\right)\right).Math.\left(I-B_f.Math.B_f^{†}\right).Math.\mathrm{KC}.Math.\hat{x}\left(t\right)& \left(19\right)\end{array}$ the boundary condition of the adjoint equation (19) is
G(tf)=0  (20).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0033] FIG. 1 is a flow chart of reconfiguration control design for a bumpless virtual actuator of an aero-engine under actuator fault;

[0034] FIG. 2 is a virtual actuator switch framework of an aero-engine actuator fault system;

[0035] 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;

[0036] 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;

[0037] 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

[0038] 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

[0039] 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:

[0040] 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;

[0041] 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;

[0042] step 3: designing a virtual actuator as:

[00010]$\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{~}{x}}\left(t\right)=A.Math.\tilde{x}\left(t\right)-B_f.Math.u_w\left(t\right)\\ u_f\left(t\right)=u_w\left(t\right)-B_f^{†}.Math.B_f.Math.{\mathrm{Ky}}_c\left(t\right)\\ y_c\left(t\right)=C.Math.\tilde{x}\left(t\right)+y_f\left(t\right)\end{array}& \left(21\right)\end{array}$

[0043] 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)

[0044] 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.

[0045] step 4: designing the virtual controller as:

[00011]$\hspace{1em}\begin{array}{cc}\{\begin{array}{c}{u}_f\left(t\right)=u_w\left(t\right)-B_f^{†}.Math.B_f.Math.K.Math.y_c\left(t\right)\\ y_c\left(t\right)=C.Math.\tilde{x}\left(t\right)+y_f\left(t\right)\\ \stackrel{.}{\stackrel{~}{x}}\left(t\right)=A.Math.\tilde{x}\left(t\right)-B_f.Math.u_w\left(t\right)-\left(I-B_f.Math.B_f^{†}\right).Math.\mathrm{BKC}.Math.\hat{x}\left(t\right)\\ \stackrel{.}{\hat{x}}\left(t\right)=\left(A-\mathrm{BKC}\right).Math.\hat{x}\left(t\right)\\ \hat{x}\left(t_0\right)=x_f\left(t_0\right)+\tilde{x}\left(t_0\right)\end{array}& \left(21\right)\end{array}$

[0046] 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)

[0047] 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).

[00012]$\begin{array}{cc}-\stackrel{.}{E}\left(t\right)=E\left(t\right).Math.\left(I-{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T.Math.Q\right).Math.A+A^T\left(I-Q.Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T\right).Math.E\left(t\right)-E\left(t\right).Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T.Math.E\left(t\right)+A^T.Math.Q.Math.A-A^T.Math.Q.Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.B_f^T.Math.\mathrm{QA}& \left(23\right)\\ .Math.C^T.Math.E\left(\mathrm{tf}\right).Math.C=R& \left(24\right)\\ \stackrel{.}{G}\left(t\right)=\left(\left(A^T.Math.Q+E\left(t\right)\right).Math.{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}-A^T\right).Math.G\left(t\right)+\left(E\left(t\right)+A^T.Math.Q\right).Math..Math..Math.\left(I-{B_f\left(B_f^T\left(P+Q\right).Math.B_f\right)}^{-1}.Math.\left(P+Q\right)\right).Math.\left(I-B_f.Math.B_f^{†}\right).Math.\mathrm{KC}.Math.\hat{x}\left(t\right)& \left(25\right)\\ .Math.G\left(\mathrm{tf}\right)=0& \left(26\right)\end{array}$

[0048] 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.

[0049] 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:

[00013]$\begin{array}{cc}A=\left[\begin{array}{cc}-6.5.Math.8.Math.6.Math.5& 21.8290\\ -0.6.Math.5.Math.0.Math.4& 0.2127\end{array}\right],B=\left[\begin{array}{cc}0.0.Math.7.Math.5.Math.4& 0.2371\\ 0.262.Math.9& 0.1484\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],x\left(0\right)={\left[-2.Math.0,3.Math.5\right]}^T& \left(27\right)\end{array}$

[0050] 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.

[0051] Suppose the actuator fault occurs at t=0.5 s, B.sub.f is diagnosed at t=3 s.

[00014]$\begin{array}{cc}{B}_f=\left[\begin{array}{cc}0.6198& 0.477.Math.2\\ 0.3.Math.2.Math.3.Math.3& 0.1434\end{array}\right]& \left(28\right)\end{array}$

[0052] 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.

[0053] 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:

[00015]$\begin{array}{cc}A=\left[\begin{array}{cc}-3.855.Math.7& 1.4467\\ 0.4.Math.6.Math.9.Math.0& -4.7.Math.081\end{array}\right],B=\left[\begin{array}{c}2.Math.3.Math.0.6.Math.7.Math.3.Math.9\\ 653.554.Math.7\end{array}\right],C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],x\left(0\right)={\left[-8.Math.0,.Math.-1.Math.03.5\right]}^T& \left(27\right)\end{array}$

[0054] 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.

[0055] Suppose the actuator fault occurs at t=0.4 s, B.sub.f is diagnosed at t=0.8 s.

[00016]$\begin{array}{cc}{B}_f=\left[\begin{array}{c}161.4717\\ -5.Math.2.Math.2.8.Math.4.Math.3.Math.8\end{array}\right]& \left(28\right)\end{array}$

[0056] 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.