METHOD FOR ACTIVE FAULT TOLERANT CONTROL OF TURBOFAN ENGINE CONTROL SYSTEM

Abstract

A method for active fault tolerant control of a turbofan engine control system designs a linear parameter varying gain scheduling robust tracking controller with good dynamic performance. Adaptive estimation of fault amplitude of a sensor and an actuator is realized according to the change of the operating state of the turbofan engine to accurately reconfigure a fault signal. An active fault tolerant control strategy based on a virtual actuator is designed according to a fault estimation result. Without redesigning the controller, through the designed active fault tolerant control strategy, on the premise of ensuring the stability of the control system, a control effect similar to that of the control system without fault is obtained.

Claims

1. A method for active fault tolerant control of a turbofan engine control system, comprising the following steps: step 1: establishing a turbofan engine LPV model based on turbofan engine test experimental data: $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{x}=A\left(\lambda \right).Math.x+B\left(\lambda \right).Math.u_c+\mathrm{Ed}\\ y=\mathrm{Cx}+\mathrm{Gd}\end{array}& \left(1\right)\end{array}$ wherein x∈R.sup.n is a state variable u.sub.c∈R.sup.m is the control input of a turbofan engine; d∈R.sup.q is a disturbance signal: output is y∈R.sup.p; the value of a scheduling parameter λ is a normalized relative conversion speed of high pressure rotors of the turbofan engine; λ.sub.min≤λ≤λ.sub.max; λ.sub.min and λ.sub.max are respectively a minimum value and a maximum value of the scheduling parameter; system matrices are A(λ)∈R.sup.n×n, B(λ)∈R.sup.n×m, C∈R.sup.n×m, E∈R.sup.n×q, G∈R.sup.p×q; R.sup.(⋅) represents a (⋅)-dimensional real column vector; and R.sup.a×b represents a a×b-dimensional real matrix; step 2: designing a LPV gain scheduling robust tracking controller for the turbofan engine LPV model with disturbance; step 2.1: introducing a new state variable x.sub.e, defined as
x.sub.e=∫.sub.0.sup.ie(s)ds=∫.sub.0.sup.i(y(s)−y.sub.r(s))ds  (2) wherein e(⋅) is a tracking error and y.sub.r(⋅) is a desired tracking signal: rewriting formula (1) into a formula (3) of a generalized form $\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\stackrel{\_}{x}}=\stackrel{\_}{A}\left(\lambda \right).Math.\stackrel{\_}{x}+{\stackrel{\_}{B}}_1.Math.\stackrel{\_}{w}+{\stackrel{\_}{B}}_2\left(\lambda \right).Math.u_c\\ \stackrel{\_}{z}={\stackrel{\_}{C}}_1.Math.\stackrel{\_}{x}+{\stackrel{\_}{D}}_{11}.Math.\stackrel{\_}{w}\end{array}.Math..Math.\mathrm{wherein}.Math..Math.\stackrel{\_}{x}=\left[\begin{array}{c}x\\ x_e\end{array}\right],.Math.\stackrel{\_}{w}=\left[\begin{array}{c}d\\ y_r\end{array}\right],\stackrel{\_}{z}=\left[\begin{array}{c}e\\ x_e\end{array}\right],\stackrel{\_}{A}\left(\lambda \right)=\left[\begin{array}{cc}A\left(\lambda \right)& 0\\ C& 0\end{array}\right],{\stackrel{\_}{B}}_1=\left[\begin{array}{cc}E& 0\\ G& -I\end{array}\right],.Math.{\stackrel{\_}{B}}_2\left(\lambda \right)=\left[\begin{array}{c}B\left(\lambda \right)\\ 0\end{array}\right],{\stackrel{\_}{C}}_1=\left[\begin{array}{cc}C& 0\\ 0& I\end{array}\right].Math..Math.\mathrm{and}.Math..Math.{\stackrel{\_}{D}}_{11}=\left[\begin{array}{cc}G& -I\\ 0& 0\end{array}\right];& \left(3\right)\end{array}$ step 2.2: for the formula (3), constructing and solving the following linear matrix inequalities (LMIs): $\begin{array}{cc}\left[\begin{array}{ccc}\left({\stackrel{\_}{A}}_i.Math.X+{\stackrel{\_}{B}}_{2.Math.i}.Math.V_i\right)+{\left({\stackrel{\_}{A}}_i.Math.X+{\stackrel{\_}{B}}_{2.Math.i}.Math.V_i\right)}^T& {\stackrel{\_}{B}.Math.}_1& {\left({\stackrel{\_}{C}}_1.Math.X\right)}^T\\ 0& -\gamma .Math..Math.I& {\stackrel{\_}{D}.Math.}_{11}^T\\ 0& 0& -\gamma .Math..Math.I\end{array}\right]& \left(4\right)\end{array}$ wherein i=1, 2, Ā.sub.2=Ā(λ.sub.min), Ā.sub.2=Ā(λ.sub.max) B.sub.21=B.sub.2(λ.sub.min) and B.sub.22=B.sub.2(λ.sub.max); γ is a desired value of H.sub.∞ norm of a close loop transfer function T.sub.wz(s) in the generalized form (3); I is an identity matrix; and the formula (4) is solved to obtain matrices X and V.sub.i; step 2.3: computing the output of the LPV gain scheduling robust tracking controller: $\begin{array}{cc}{u}_c=K\left(\lambda \right).Math.x=\underset{i=1}{\overset{2}{.Math.}}.Math.{\alpha }_i.Math.V_i.Math.X^{-1}.Math.x.Math..Math.\mathrm{wherein}.Math..Math.K\left(\lambda \right)=\underset{i=1}{\overset{2}{.Math.}}.Math.{\alpha }_i.Math.V_i.Math.X^{-1},{\alpha }_1\left(\lambda \right)=\frac{{\lambda }_{\max }-\lambda }{{\lambda }_{\max }-{\lambda }_{\min }}.Math..Math.\mathrm{and}.Math..Math.{\alpha }_2\left(\lambda \right)=\frac{\lambda -{\lambda }_{\min }}{{\lambda }_{\max }-{\lambda }_{\min }};& \left(5\right)\end{array}$ step 3: for the turbofan engine LPV model with disturbance and sensor and actuator faults, establishing an adaptive fault estimator of the turbofan engine based on a robust H.sub.∞ optimization method to realize fault estimation of a sensor and an actuator; step 3.1: considering that the turbofan engine control system has the actuator and sensor faults, and expressing a system with fault as shown in formula (6): $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{x}}_f=A\left(\lambda \right).Math.x_f+B_f\left(\lambda \right).Math.u+\mathrm{Ed}+F_f\left(\lambda \right).Math.f\\ y_f={\mathrm{Cx}}_f+\mathrm{Gd}+H_f\left(\lambda \right).Math.f\end{array}& \left(6\right)\end{array}$ wherein x∈R.sup.n is a state variable of the system with fault u∈R.sup.m is control input of the system with fault; y.sub.f∈R.sup.p is measurement output of the system with fault; f=[f.sub.a.sup.T f.sub.s.sup.T].sup.T∈R.sup.l is a fault signal; f.sub.a∈R.sup.l.sup.1 is an actuator fault; f.sub.s∈R.sup.l.sup.2 is a sensor fault; B.sub.f(λ)∈R.sup.n×m is a matrix of the system with fault; and F.sub.f(λ)∈R.sup.n×l and H.sub.f(λ)∈R.sup.p×l are respectively fault matrices of the actuator and the sensor; step 3.2: separating a time varying part from a time invariant part in the formula (6), and rewriting into the following form $\begin{array}{cc}\hspace{1em}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_f\\ z_{\lambda }\\ y_f\end{array}\right]=\left[\begin{array}{ccc}A& B_{f.Math..Math.1}& B_{f.Math..Math.2}\\ C_{f.Math..Math.1}& D_{f.Math..Math.11}& D_{f.Math..Math.12}\\ C_{f.Math..Math.2}& D_{f.Math..Math.21}& D_{f.Math..Math.22}\end{array}\right]\left[\begin{array}{c}{x}_f\\ w_{\lambda }\\ w\end{array}\right]\\ w_{\lambda }=\Lambda .Math..Math.z_{\lambda }\end{array}& \left(7\right)\end{array}$ wherein external input is w=[u.sup.T d.sup.T f.sup.T].sup.T; z.sub.λ, w.sub.λ∈R.sup.r are respectively input and output variables of a r-dimensional time varying subsystem Λ=λI in the formula (6): B.sub.f1∈R.sup.n×r, B.sub.f2∈R.sup.n×(m+q+l), C.sub.f1∈R.sup.r×n, C.sub.f2∈R.sup.p×n, D.sub.f11∈R.sup.r×r D.sub.12∈R.sup.r×(m+q+l), D.sub.f21∈∈R.sup.p×r and D.sub.f22∈R.sup.p×(m+q+l) are system state space matrices; based on the formula (7), constructing a fault estimator state space expression as follows: $\begin{array}{cc}\hspace{1em}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_e\\ \hat{f}\\ z_{e.Math..Math.\lambda }\end{array}\right]=\left[\begin{array}{ccc}{A}_e& B_{e.Math..Math.1}& B_{e.Math..Math.2}\\ C_{e.Math..Math.1}& D_{e.Math..Math.11}& D_{e.Math..Math.12}\\ C_{e.Math..Math.2}& D_{e.Math..Math.21}& D_{e.Math..Math.22}\end{array}\right]\left[\begin{array}{c}{x}_e\\ u_e\\ w_{e.Math..Math.\lambda }\end{array}\right]\\ w_{e.Math..Math.\lambda }=\Lambda .Math..Math.z_{e.Math..Math.\lambda }\end{array}& \left(8\right)\end{array}$ wherein x.sub.e∈R.sup.k, u.sub.e=[u.sup.T Y.sub.f.sup.T].sup.T∈R.sup.(p+m) and {circumflex over (f)}∈R.sup.t respectively represent a state variable, a control input and a fault estimation output of the fault estimator; z.sub.eλ∈R.sup.r and w.sub.eλ∈R.sup.r respectively represent an input and an output of the time varying part of the fault estimator; A.sub.e∈R.sup.k×k, B.sub.e1∈R.sup.k×(m+p), B.sub.e2∈R.sup.k×r, C.sub.e1∈R.sup.l×k, C.sub.e2∈R.sup.r×k, D.sub.e11∈R.sup.l×(p+m), D.sub.e12∈R.sup.l×r, D.sub.e21∈R.sup.r×(p+m) and D.sub.e22∈R.sup.r×r are fault estimator coefficient matrices to be designed; step 3.3: constructing a state space joint representation of the formula (7) of the system with fault of the turbofan engine and the formula (8) of the fault estimator: $\hspace{1em}\begin{array}{cc}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_f\\ \frac{{\stackrel{.}{x}}_e}{z_{e.Math..Math.\lambda }}\\ \frac{z_{\lambda }}{e_f}\end{array}\right]=\left[\begin{array}{ccc}\stackrel{\_.Math.}{A}& {\stackrel{\_.Math.}{B}}_1& {\stackrel{\_.Math.}{B}}_2\\ {\stackrel{\_}{C}}_1& {\stackrel{\_.Math.}{D}}_{11}& {\stackrel{\_.Math.}{D}}_{12}\\ {\stackrel{\_}{C}}_2& {\stackrel{\_.Math.}{D}}_{21}& {\stackrel{\_.Math.}{D}}_{22}\end{array}\right]\left[\begin{array}{c}{x}_f\\ \frac{x_e}{w_{e.Math..Math.\lambda }}\\ \frac{w_{\lambda }}{w}\end{array}\right]\\ \left[\begin{array}{c}{w}_{\lambda }\\ w_{e.Math..Math.\lambda }\end{array}\right]=\left[\begin{array}{cc}\Lambda & 0\\ 0& \Lambda \end{array}\right]\left[\begin{array}{c}{z}_{\lambda }\\ z_{e.Math..Math.\lambda }\end{array}\right]\end{array}& \left(9\right)\end{array}$ wherein a fault estimate error is e.sub.f={circumflex over (f)}−f, $\begin{array}{cc}\left[\begin{array}{ccc}\stackrel{\_.Math.}{A}& {\stackrel{\_}{B}.Math.}_1& {\stackrel{\_.Math.}{B}}_2\\ {\stackrel{\_}{C}}_1& {\stackrel{\_.Math.}{D}}_{11}& {\stackrel{\_.Math.}{D}}_{12}\\ {\stackrel{\_}{C}}_2& {\stackrel{\_.Math.}{D}}_{21}& {\stackrel{\_.Math.}{D}}_{22}\end{array}\right]=\left[\begin{array}{ccc}{A}_0& B_{01}& B_{02}\\ C_{01}& D_{01}& D_{02}\\ C_{02}& D_{03}& D_{04}\end{array}\right]+\left[\begin{array}{c}{T}_1\\ T_5\\ T_6\end{array}\right].Math.\Gamma \left[\begin{array}{ccc}{T}_2& T_3& T_4\end{array}\right].Math..Math..Math.A_0=\left[\begin{array}{cc}A& 0\\ 0& 0\end{array}\right],B_{01}=\left[\begin{array}{cc}0& B_{f.Math..Math.1}\\ 0& 0\end{array}\right],B_{02}=\left[\begin{array}{c}{B}_{f.Math..Math.2}\\ 0\end{array}\right]& \left(10\right)\\ .Math.C_{01}=\left[\begin{array}{cc}0& 0\\ C_{f.Math..Math.1}& 0\end{array}\right],D_{01}=\left[\begin{array}{cc}0& 0\\ 0& D_{f.Math..Math.11}\end{array}\right],D_{02}=\left[\begin{array}{c}0\\ D_{f.Math..Math.12}\end{array}\right].Math..Math..Math.C_{02}=\left[\begin{array}{cc}0& 0\end{array}\right],D_{03}=\left[\begin{array}{cc}0& 0\end{array}\right],D_{04}=D_{22}& \left(11\right)\\ .Math.T_1=\left[\begin{array}{ccc}0& 0& 0\\ I& 0& 0\end{array}\right],T_2=\left[\begin{array}{cc}0& I\\ C_3& 0\\ 0& 0\end{array}\right],T_3=\left[\begin{array}{cc}0& 0\\ 0& D_{31}\\ I& 0\end{array}\right].Math..Math..Math.T_4=\left[\begin{array}{c}0\\ D_{32}\\ 0\end{array}\right],T_5=\left[\begin{array}{ccc}0& 0& I\\ 0& D_{13}& 0\end{array}\right],T_6=\left[\begin{array}{ccc}0& D_{23}& 0\end{array}\right]& \left(12\right)\\ .Math.\Gamma =\left[\begin{array}{ccc}{A}_e& B_{e.Math..Math.1}& B_{e.Math..Math.2}\\ C_{e.Math..Math.1}& D_{e.Math..Math.11}& D_{e.Math..Math.12}\\ C_{e.Math..Math.2}& D_{e.Math..Math.21}& D_{e.Math..Math.22}\end{array}\right].Math..Math.C_3=\left[\begin{array}{c}0\\ C_{f.Math..Math.2}\end{array}\right],D_{22}=\left[\begin{array}{cc}0& -I\\ 0& 0\end{array}\right],.Math..Math.D_{23}=\left[\begin{array}{c}I\\ 0\end{array}\right],D_{31}=\left[\begin{array}{c}0\\ D_{f.Math..Math.21}\end{array}\right],D_{32}=\left[\begin{array}{ccc}I& 0& 0\\ & D_{f.Math..Math.22}& \end{array}\right]& \left(13\right)\end{array}$ and Γ are estimation matrices of the fault estimator; step 3.4: setting $X=\left[\begin{array}{cc}L& V\\ V^T& Y\end{array}\right],X^{-1}=\left[\begin{array}{cc}J& W\\ W^T& Z\end{array}\right]$ wherein L, V and Y respectively represent sub-block matrices of X; and J, W and Z respectively represent sub-block matrices of X.sup.−1; constructing a matrix P and an inverse matrix {tilde over (P)} as shown in formula (14): $\begin{array}{cc}P=\left[\begin{array}{cc}Q& S\\ S^T& R\end{array}\right]=\left[\begin{array}{cccc}{Q}_1& Q_2& S_1& S_2\\ Q_2^T& Q_3& S_3& S_4\\ S_1^T& S_3^T& R_1& R_2\\ S_2^T& S_4^T& R_2^T& R_3\end{array}\right],.Math.P^{-1}=\tilde{P}=\left[\begin{array}{cc}\tilde{Q}& \tilde{S}\\ {\tilde{S}}^T& \tilde{R}\end{array}\right]=\left[\begin{array}{cccc}{\tilde{Q}}_1& {\tilde{Q}}_2& {\tilde{S}}_1& {\tilde{S}}_2\\ {\tilde{Q}}_2^T& {\tilde{Q}}_3& {\tilde{S}}_3& {\tilde{S}}_4\\ {\tilde{S}}_1^T& {\tilde{S}}_3^T& {\tilde{R}}_1& {\tilde{R}}_2\\ {\tilde{S}}_2^T& {\tilde{S}}_4^T& {\tilde{R}}_2^T& {\tilde{R}}_3\end{array}\right]& \left(14\right)\end{array}$ wherein Q.sub.1, Q.sub.2 and Q.sub.3 respectively represent sub-block matrices of Q; S.sub.1, S.sub.2, S.sub.3 and S.sub.4 respectively represent sub-block matrices of S; R.sub.1, R.sub.2 and R.sub.3 respectively represent sub-block matrices of R; {tilde over (Q)}, {tilde over (S)} and {tilde over (R)} respectively represent sub-block matrices of {tilde over (P)}; {tilde over (Q)}.sub.1, {tilde over (Q)}.sub.2 and {tilde over (Q)}.sub.3 respectively represent sub-block matrices of {tilde over (Q)}; {tilde over (S)}.sub.1, {tilde over (S)}.sub.2, {tilde over (S)}.sub.3 and {tilde over (S)}.sub.4 respectively represent sub-block matrices of {tilde over (S)}; {tilde over (R)}.sub.1, {tilde over (R)}.sub.2 and {tilde over (R)}.sub.3 respectively represent sub-block matrices of {tilde over (R)}; constructing the following LMIs, and combining to solve corresponding matrix solutions L, J, Q.sub.3, R.sub.3, S.sub.4, {tilde over (Q)}.sub.3, {tilde over (R)}.sub.3 and {tilde over (S)}.sub.4: $\begin{array}{cc}{N_L^T\left[\begin{array}{ccc}I& 0& 0\\ 0& I& 0\\ 0& 0& I\\ A& B_{f.Math..Math.1}& B_{f.Math..Math.2}\\ C_{f.Math..Math.1}& D_{f.Math..Math.11}& D_{f.Math..Math.12}\\ 0& 0& D_{22}\end{array}\right]}^T\left[\begin{array}{cccccc}0& 0& 0& L& 0& 0\\ 0& Q_3& 0& 0& S_4& 0\\ 0& 0& -\gamma .Math..Math.I& 0& 0& 0\\ L& 0& 0& 0& 0& 0\\ 0& S_4^T& 0& 0& R_3& 0\\ 0& 0& 0& 0& 0& {\gamma }^{-1}.Math.I\end{array}\right].Math.\hspace{1em}\left[\begin{array}{ccc}I& 0& 0\\ 0& I& 0\\ 0& 0& I\\ A& B_{f.Math..Math.1}& B_{f.Math..Math.2}\\ C_{f.Math..Math.1}& D_{f.Math..Math.11}& D_{f.Math..Math.12}\\ 0& 0& D_{22}\end{array}\right].Math.N_L<0& \left(15\right)\\ {N_J^T\left[\begin{array}{ccc}-A^T& -C_{f.Math..Math.1}^T& 0\\ -B_{f.Math..Math.1}^T& -D_{f.Math..Math.11}^T& 0\\ -B_{f.Math..Math.2}^T& -D_{f.Math..Math.12}^T& -D_{22}^T\\ I& 0& 0\\ 0& I& 0\\ 0& 0& I\end{array}\right]}^T\left[\begin{array}{cccccc}0& 0& 0& J& 0& 0\\ 0& {\tilde{Q}}_3& 0& 0& {\tilde{S}}_4& 0\\ 0& 0& -{\gamma }^{-1}.Math..Math.I& 0& 0& 0\\ J& 0& 0& 0& 0& 0\\ 0& {\tilde{S}}_4^T& 0& 0& {\tilde{R}}_3& 0\\ 0& 0& 0& 0& 0& \gamma .Math..Math.I\end{array}\right].Math.\hspace{1em}\left[\begin{array}{ccc}-A^T& -C_{f.Math..Math.1}^T& 0\\ -B_{f.Math..Math.1}^T& -D_{f.Math..Math.11}^T& 0\\ -B_{f.Math..Math.2}^T& -D_{f.Math..Math.12}^T& -D_{22}^T\\ I& 0& 0\\ 0& I& 0\\ 0& 0& I\end{array}\right].Math.N_J>0& \left(16\right)\\ .Math.\left[\begin{array}{cc}J& I\\ I& L\end{array}\right]>0& \left(17\right)\\ .Math.R>0,Q=-R,S+S^T=0& \left(18\right)\end{array}$ wherein N.sub.L and N.sub.J are respectively the bases of the nuclear spaces of [C.sub.3 D.sub.31 D.sub.32] and [0 D.sub.13.sup.T D.sub.23.sup.T]; step 3.5: further, solving X in the formula (17) according to a solving result of the step 3.4; $\begin{array}{cc}X\left[\begin{array}{cc}J& I\\ W^T& 0\end{array}\right]=\left[\begin{array}{cc}I& L\\ 0& V^T\end{array}\right]& \left(17\right)\end{array}$ solving P according to P{tilde over (P)}=I; step 3.6: solving the following LMIs to obtain an estimation matrix Γ of the fault estimator: $\begin{array}{cc}.Math.\Psi +{\stackrel{\_}{P}}^T.Math.{\Gamma }^T.Math.{\stackrel{\_}{Q}}_X+{\stackrel{\_}{Q}}_X^T.Math.\Gamma .Math..Math.\stackrel{\_}{P}<0.Math..Math.\mathrm{wherein}.Math..Math.\Psi =[.Math.\begin{array}{ccccc}{A}_0^T.Math.X+{\mathrm{XA}}_0& {\mathrm{XB}}_{01}+C_{01}^T.Math.S^T& {\mathrm{XB}}_{02}& C_{01}^T& C_{02}^T\\ {\mathrm{SC}}_{01}+B_{01}^T.Math.X^T& Q+D_{01}^T.Math.S^T+{\mathrm{SD}}_{01}& {\mathrm{SD}}_{02}& D_{01}^T& D_{03}^T\\ B_{02}^T.Math.X^T& D_{02}^T.Math.S^T& -\gamma .Math..Math.I& D_{02}^T& D_{04}^T\\ C_1& D_{01}& D_{02}& -\tilde{R}& 0\\ C_{02}& D_{03}& D_4& 0& -\gamma .Math..Math.I\end{array}].Math..Math..Math.\stackrel{\_}{P}=\left[\begin{array}{ccccc}{T}_2& T_3& T_4& 0& 0\end{array}\right],{\stackrel{\_}{Q}}_X=\left[\begin{array}{ccccc}{T}_1^T.Math.X& T_5^T.Math.S^T& 0& T_5^T& T_6^T\end{array}\right]& \left(19\right)\end{array}$ further, computing a coefficient matrix of the fault estimator: $\begin{array}{cc}\left[\begin{array}{cc}{A}_E\left(\lambda \right)& B_E\left(\lambda \right)\\ C_E\left(\lambda \right)& D_E\left(\lambda \right)\end{array}\right]=\left[\begin{array}{cc}{A}_e& B_{e.Math..Math.1}\\ C_{e.Math..Math.1}& D_{e.Math..Math.11}\end{array}\right]+\hspace{1em}\left[\begin{array}{c}{B}_{e.Math..Math.2}\\ D_{e.Math..Math.12}\end{array}\right].Math.{\Lambda \left(I-D_{e.Math..Math.22}.Math.\Lambda \right)}^{-1}\left[\begin{array}{cc}{C}_{e.Math..Math.2}& D_{e.Math..Math.21}\end{array}\right]& \left(20\right)\end{array}$ step 4: designing an active fault tolerant controller of a turbofan engine based on a virtual actuator according to a fault estimation result; and without redesigning the controller, making the control system stable and obtaining a control effect similar to that of the system without fault; step 4.1: considering the system with fault of the turbofan engine; and when the sensor and actuator faults exist, designing the virtual actuator based on a reconfiguration principle, with a state space model representation of a reconfigured system as follows: $\begin{array}{cc}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{\stackrel{\_}{x}}}_f\\ {\stackrel{.}{x}}_{\Delta }\end{array}\right]=\left[\begin{array}{cc}\stackrel{\_}{A}\left(\lambda \right)& {\stackrel{\_}{B}}_{2.Math..Math.f}\left(\lambda \right).Math.C_{\Delta }\left(\lambda \right)\\ 0& A_{\Delta }\left(\lambda \right)\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{x}}_f\\ x_{\Delta }\end{array}\right]+\left[\begin{array}{c}{\stackrel{\_}{B}}_1\\ 0\end{array}\right].Math.\stackrel{\_}{w}+\left[\begin{array}{c}{\stackrel{\_}{B}}_{2.Math.f}\left(\lambda \right).Math.D_{\Delta }\left(\lambda \right)\\ B_{\Delta }\left(\lambda \right)\end{array}\right].Math.u_c\\ z_{\mathrm{re}}=\left[\begin{array}{cc}{\stackrel{\_}{C}}_1& 0\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{x}}_f\\ x_{\Delta }\end{array}\right]+{\stackrel{\_}{D}}_{11}.Math.\stackrel{\_}{w}\\ x_c=\left[\begin{array}{cc}I& I\end{array}\right]\left[\begin{array}{c}{\stackrel{\_}{x}}_f\\ x_{\Delta }\end{array}\right]\end{array}.Math.\mathrm{wherein}.Math..Math.x_f=\left[\begin{array}{c}{x}_f\\ x_{\mathrm{ef}}\end{array}\right];x_{\mathrm{ef}}={\int }_0^t.Math.e_f\left(s\right).Math.\mathrm{ds}={\int }_0^t.Math.\left(y_f\left(s\right)-y_r\left(s\right)\right).Math.\mathrm{ds};& \left(21\right)\end{array}$ x.sub.Δ is a state variable of the virtual actuator; ${\stackrel{\_}{B}}_{2.Math.f}\left(\lambda \right)=\left[\begin{array}{c}{B}_f\left(\lambda \right)\\ 0\end{array}\right];A_{\Delta }\left(\lambda \right)=\stackrel{\_}{A}\left(\lambda \right)-{\stackrel{\_}{B}}_{2.Math.f}\left(\lambda \right).Math.M\left(\lambda \right);$ $B_{\Delta }\left(\lambda \right)=B_2\left(\lambda \right)-{\stackrel{\_}{B}}_{2.Math.f}\left(\lambda \right).Math.N\left(\lambda \right);\stackrel{\_}{w}=\left[\begin{array}{c}d\\ y_r\end{array}\right];C_{\Delta }\left(\lambda \right)=M\left(\lambda \right);$ $D_{\Delta }\left(\lambda \right)=N\left(\lambda \right);{\stackrel{\_}{C}}_1=\left[\begin{array}{cc}C& 0\\ 0& I\end{array}\right];{\stackrel{\_}{D}}_{11}=\left[\begin{array}{cc}G& -I\\ 0& 0\end{array}\right],x_c=x_{\Delta }+{\stackrel{\_}{x}}_f;$ z.sub.re is a controlled output of the reconfigured system; M(λ) and N(λ) are to-be-solved matrices in an active fault tolerant control law; solving positive definite matrices X.sub.v, Y.sub.1 and Y.sub.2 according to LMIs combined by (22)-(24); $\begin{array}{cc}{\stackrel{\_}{A}}_i.Math.X_v-{\stackrel{\_}{B}}_{2.Math.f_i}.Math.Y_i+X_v.Math.{\stackrel{\_}{A}}_i^T-Y_i^T.Math.{\stackrel{\_}{B}}_{2.Math.f_i}^T+2.Math..Math.\rho .Math..Math.X_v<0& \left(22\right)\\ \left[\begin{array}{cc}-{\mathrm{rX}}_v& {\mathrm{qX}}_c+{\stackrel{\_}{A}}_i.Math.X_v-{\stackrel{\_}{B}}_{2.Math.f_i}.Math.Y_i\\ *& -{\mathrm{rX}}_v\end{array}\right]<0& \left(23\right)\\ \left[\begin{array}{cc}\sin \left(\theta \right).Math.\left(\begin{array}{c}{\stackrel{\_}{A}}_i.Math.X_v-{\stackrel{\_}{B}}_{2.Math.f_i}.Math.Y_i+\\ X_v.Math.{\stackrel{\_}{A}}_i^T-Y_i^T.Math.{\stackrel{\_}{B}}_{2.Math.f_i}^T\end{array}\right)& \cos \left(\theta \right).Math.\left(\begin{array}{c}{\stackrel{\_}{A}}_i.Math.X_v-{\stackrel{\_}{B}}_{2.Math.f_i}.Math.Y_i-\\ X_v.Math.{\stackrel{\_}{A}}_i^T-Y_i^T.Math.{\stackrel{\_}{B}}_{2.Math.f_i}^T\end{array}\right)\\ \cos \left(\theta \right).Math.\left(\begin{array}{c}{X}_v^T.Math.{\stackrel{\_}{A}}_i^T-Y_i.Math.{\stackrel{\_}{B}}_{2.Math.f_i}^T-\\ {\stackrel{\_}{A}}_i.Math.X_v^T-{\stackrel{\_}{B}}_{2.Math.f_i}.Math.Y_i\end{array}\right)& \sin \left(\theta \right).Math.\left(\begin{array}{c}{\stackrel{\_}{A}}_i.Math.X_v-{\stackrel{\_}{B}}_{2.Math.f_i}.Math.Y_i+\\ X_v.Math.{\stackrel{\_}{A}}_i^T-Y_i^T.Math.{\stackrel{\_}{B}}_{2.Math.f_i}^T\end{array}\right)\end{array}\right]<0& \left(24\right)\end{array}$ wherein i=1,2; ρ is a minimum decay rate of an LMI region; r is a radius of the LMI region; q is a center of a circle; θ is an intersection angle of close loop poles and a transverse axis in the LMI region; step 4.2: obtaining a matrix M.sub.i according to Y.sub.i=M.sub.iX.sub.v; step 4.3: computing $N\left(\lambda \right)=\underset{i=1}{\overset{2}{.Math.}}.Math.{\alpha }_i.Math.N_i=\underset{i=1}{\overset{2}{.Math.}}.Math.{\alpha }_i.Math.{\stackrel{\_}{B}}_{2.Math.f_i}^{+}.Math.{\stackrel{\_}{B}}_{2_i},$ wherein B.sub.2f.sub.i.sup.+ represents pseudo inverse of B.sub.2f.sub.i; step 4.4: computing a system matrix;
A.sub.Δ(λ)=Ā(λ)−B.sub.2f(λ)M(λ), B.sub.Δ(λ)=B.sub.2(λ)−B.sub.2f(λ)
C.sub.Δ(λ)=M(λ), D.sub.Δ(λ)=N(λ) constructing a state space equation and a control law of the active fault tolerant controller: $\begin{array}{cc}\{\begin{array}{c}{\stackrel{.}{x}}_{\Delta }=A_{\Delta }\left(\lambda \right).Math.x_{\Delta }+B_{\Delta }\left(\lambda \right).Math.u_c\\ u_f=C_{\Delta }\left(\lambda \right).Math.x_{\Delta }+D_{\Delta }\left(\lambda \right).Math.u_c-{\stackrel{\_}{B}}_{2.Math.f}^{-1}.Math.\stackrel{\_}{F}\left(\lambda \right).Math.\hat{f}\end{array}.& \left(25\right)\end{array}$

Description

DESCRIPTION OF DRAWINGS

[0030] FIG. 1 is a design flow chart of an active fault tolerant controller of a sensor and an actuator of a turbofan engine control system;

[0031] FIG. 2 is a block diagram of an LPV gain scheduling robust tracking controller of a turbofan engine;

[0032] FIG. 3(a) shows turbofan engine LPV gain scheduling robust tracking control simulation results when relative conversion speed of high pressure rotors is 88%;

[0033] FIG. 3(b) shows turbofan engine LPV gain scheduling robust tracking control simulation results when relative conversion speed of high pressure rotors is 94%;

[0034] FIG. 4 is a block diagram of a fault estimator of a turbofan engine;

[0035] FIG. 5(a) shows fault estimation results of a sensor of a turbofan engine;

[0036] FIG. 5(b) shows fault estimation results of an actuator of a turbofan engine;

[0037] FIG. 6 is a block diagram of an active fault tolerant controller of a sensor and an actuator of a turbofan engine control system;

[0038] FIG. 7(a) shows output of a system with fault when the relative conversion speed of high pressure rotors is 90%;

[0039] FIG. 7(b) shows output of a system with fault when the relative conversion speed of high pressure rotors is 94%;

[0040] FIG. 8(a) shows normal output results of a turbofan engine control system when the relative conversion speed of high pressure rotors is 90%;

[0041] FIG. 8(b) shows active fault tolerant control results of a turbofan engine control system when the relative conversion speed of high pressure rotors is 90%;

[0042] FIG. 9(a) shows normal output results of a turbofan engine control system when the relative conversion speed of high pressure rotors is 94%; and

[0043] FIG. 9(b) shows active fault tolerant control results of a turbofan engine control system when the relative conversion speed of high pressure rotors is 94%.

DETAILED DESCRIPTION

[0044] The present invention is further described below in combination with the drawings. The research object of the present invention is the turbofan engine after the sensor and actuator faults of the control system. The design method is shown in a flow chart of FIG. 1. Detailed design steps are as follows.

[0045] Step 1: establishing a turbofan engine LPV model based on turbofan engine test experimental data:

[00021]$\begin{array}{cc}.Math.\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\end{array}\right]=\left[\begin{array}{cc}{a}_{11}\left(\lambda \right)& a_{12}\left(\lambda \right)\\ a_{21}\left(\lambda \right)& a_{22}\left(\lambda \right)\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}{b}_1\left(\lambda \right)\\ b_2\left(\lambda \right)\end{array}\right].Math.u+\left[\begin{array}{c}0.1\\ 0\end{array}\right].Math.d\\ y\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+0.2.Math.d\end{array}& \\ .Math.\mathrm{wherein}& \\ \left[\begin{array}{cc}{a}_{11}\left(\lambda \right)& a_{12}\left(\lambda \right)\\ a_{21}\left(\lambda \right)& a_{22}\left(\lambda \right)\end{array}\right]=\left[\begin{array}{cc}-2.6748& 0.6877\\ 1.0704& -4.4672\end{array}\right]+\lambda \left[\begin{array}{cc}0.5199& -2.4061\\ 0.1049& -0.8365\end{array}\right]& \left(26\right)\\ .Math.\left[\begin{array}{c}{b}_1\left(\lambda \right)\\ b_2\left(\lambda \right)\end{array}\right]=\left[\begin{array}{c}0.0033\\ 0.0012\end{array}\right]+\lambda \left[\begin{array}{c}-0.0004\\ -0.0001\end{array}\right]& \left(27\right)\end{array}$

a scheduling parameter λ is a normalized relative conversion speed of high pressure rotors of the turbofan engine: −1≤λ≤1; thus, a time varying parameter k forms a parametric polyhedron with −1 and 1 as vertexes, and the disturbance d takes Gaussian white noise with a standard deviation of 0.0001.

[0046] Step 2: as shown in FIG. 2, designing a LPV gain scheduling robust tracking controller for the turbofan engine LPV model with disturbance.

[0047] Setting γ=2 and solving LMI (4) to obtain the following corresponding matrix solutions X and V.sub.i (i=1, 2).

[00022]$X=\left[\begin{array}{ccc}10.3036& 1.4883& -1.7799\\ 1.4883& 1.5432& -0.9087\\ -1.7799& -0.9087& 0.8419\end{array}\right]$$V_1=\left[\begin{array}{ccc}5589& -3460& -946\end{array}\right]$$V_2=\left[\begin{array}{ccc}5894& -1975& -2096\end{array}\right]$

Obtaining the gain K(k) of the speed tracking controller in combination with the formula (5), and

[00023]${\alpha }_1\left(\lambda \right)=\frac{1-\lambda }{1-\left(-1\right)},{\alpha }_2\left(\lambda \right)=\frac{\lambda -\left(-1\right)}{1-\left(-1\right)}$

[0048] Thus, the gain of the speed tracking controller can be changed with the change of the time varying parameter λ and has scheduling characteristics. FIG. 3 respectively shows turbofan engine LPV gain scheduling robust tracking control simulation results when the relative conversion speeds of high pressure rotors are respectively 90% and 94%. It can be seen from the simulation results that, regardless of the parameters at the vertex or the parameters at non-vertex, the designed LPV speed tracking controller can ensure that the output responds quickly and tracks a reference instruction. Therefore, the controller has good control performance within all time varying parameters, can stably and quickly realize tracking and has good robustness and stability.

[0049] Step 3: establishing an adaptive fault estimator of the turbofan engine, as shown in FIG. 4, to realize fault estimation of a sensor and an actuator: considering that the turbofan engine control system has the actuator and sensor faults, as shown in formula (28):

[00024]$\begin{array}{cc}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_1\\ {\stackrel{.}{x}}_2\end{array}\right]=\left[\begin{array}{cc}{a}_{11}\left(\lambda \right)& a_{12}\left(\lambda \right)\\ a_{21}\left(\lambda \right)& a_{22}\left(\lambda \right)\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}{b}_1\left(\lambda \right)\\ b_2\left(\lambda \right)\end{array}\right].Math.u+\left[\begin{array}{c}0.1\\ 0\end{array}\right].Math.d+\left[\begin{array}{c}{f}_1\left(\lambda \right)\\ 0\end{array}\right].Math.f\\ y=\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+0.2.Math.d+\left[\begin{array}{c}0\\ h\left(\lambda \right)\end{array}\right].Math.f\end{array}& \left(28\right)\end{array}$

[0050] wherein f.sub.1(λ)=0.1+0.01λ, −1≤λ≤1, h(λ)=0.5+0.02λ, −1≤λ≤1, and the disturbance takes Gaussian white noise with a standard deviation of 0.0001. The time varying parameter λ is real-time measurable, and the change thereof is supposed as follows

[00025]$\lambda \left(t\right)=\{\begin{array}{cc}0,& 0\le t<5.Math.s\\ -0.3333,& 5.Math.s\le t<10.Math.s\\ 0.3333,& 10.Math.s\le t<15.Math.s\\ -0.5,& 15.Math.s\le t<20.Math.s\\ -0.6667,& 20.Math.s\le t<25.Math.s\end{array}$

A multiplicative actuator fault is expr as

[00026]$\left[\begin{array}{c}{b}_{1.Math.f}\left(\lambda \right)\\ b_{2.Math.f}\left(\lambda \right)\end{array}\right]=\left[\begin{array}{c}0.1.Math.b_1\left(\lambda \right)\\ 0.1.Math.b_2\left(\lambda \right)\end{array}\right]$

[0051] A time varying part and a time invariant part in the formula (28) are separated and rewritten into the following form

[00027]$\hspace{1em}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_f\\ z_{\lambda }\\ y_f\end{array}\right]=\left[\begin{array}{ccc}A& B_{f.Math..Math.1}& B_{f.Math..Math.2}\\ C_{f.Math..Math.1}& D_{f.Math..Math.11}& D_{f.Math..Math.12}\\ C_{f.Math..Math.2}& D_{f.Math..Math.21}& D_{f.Math..Math.22}\end{array}\right]\left[\begin{array}{c}{x}_f\\ w_{\lambda }\\ w\end{array}\right]\\ w_{\lambda }=\Lambda .Math..Math.z_{\lambda }\end{array}$

wherein external input is w=[u.sup.T d.sup.T f.sup.T].sup.T; z.sub.λ, w.sub.λ∈R.sup.r are respectively input and output variables of a r-dimensional time varying subsystem Λ=λI in the formula (6): B.sub.f1∈R.sup.n×r, B.sub.f2∈R.sup.n×(m+q+l), C.sub.f1∈R.sup.r×n, C.sub.f2∈R.sup.p×n, D.sub.f11∈R.sup.r×r D.sub.12∈R.sup.r×(m+q+l), D.sub.f21∈∈R.sup.p×r and D.sub.f22∈R.sup.p×(m+q+l) are system state space matrices.

[0052] A fault estimator state space expression is constructed as follows

[00028]$\hspace{1em}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_e\\ \hat{f}\\ z_{e.Math..Math.\lambda }\end{array}\right]=\left[\begin{array}{ccc}{A}_e& B_{e.Math..Math.1}& B_{e.Math..Math.2}\\ C_{e.Math..Math.1}& D_{e.Math..Math.11}& D_{e.Math..Math.12}\\ C_{e.Math..Math.2}& D_{e.Math..Math.21}& D_{e.Math..Math.22}\end{array}\right]\left[\begin{array}{c}{x}_e\\ u_e\\ w_{e.Math..Math.\lambda }\end{array}\right]\\ w_{e.Math..Math.\lambda }=\Lambda .Math..Math.z_{e.Math..Math.\lambda }\end{array}$

wherein x.sub.e∈R.sup.k, u.sub.e=[u.sup.T Y.sub.f.sup.T].sup.T∈R.sup.(p+m) and {circumflex over (f)}∈R.sup.t respectively represent a state variable, a control input and a fault estimation output of the fault estimator; z.sub.eλ∈R.sup.r and w.sub.eλ∈R.sup.r respectively represent an input and an output of the time varying part of the fault estimator; A.sub.e∈R.sup.k×k, B.sub.e1∈R.sup.k×(m+p), B.sub.e2∈R.sup.k×r, C.sub.e1∈R.sup.l×k, C.sub.e2∈R.sup.r×k, D.sub.e11∈R.sup.l×(p+m), D.sub.e12∈R.sup.l×r, D.sub.e21∈R.sup.r×(p+m) and D.sub.e22∈R.sup.r×r are fault estimator coefficient matrices to be designed.

[0053] A state space joint representation of the system with fault of the turbofan engine and the fault estimator is constructed:

[00029]$\hspace{1em}\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{.}{x}}_f\\ \frac{{\stackrel{.}{x}}_e}{z_{e.Math..Math.\lambda }}\\ \frac{z_{\lambda }}{e_f}\end{array}\right]=\left[\begin{array}{ccc}A& B_1& B_2\\ C_1& D_{11}& D_{12}\\ C_2& D_{21}& D_{22}\end{array}\right]\left[\begin{array}{c}{x}_f\\ \frac{x_e}{w_{e.Math..Math.\lambda }}\\ \frac{w_{\lambda }}{w}\end{array}\right]\\ \left[\begin{array}{c}{w}_{\lambda }\\ w_{e.Math..Math.\lambda }\end{array}\right]=\left[\begin{array}{cc}\Lambda & 0\\ 0& \Lambda \end{array}\right]\left[\begin{array}{c}{z}_{\lambda }\\ z_{e.Math..Math.\lambda }\end{array}\right]\end{array}$

wherein a fault estimate error is e.sub.f={circumflex over (f)}−f,

[00030]$\begin{array}{cc}\left[\begin{array}{ccc}\stackrel{\_.Math.}{A}& {\stackrel{\_}{B}.Math.}_1& {\stackrel{\_.Math.}{B}}_2\\ {\stackrel{\_}{C}}_1& {\stackrel{\_.Math.}{D}}_{11}& {\stackrel{\_.Math.}{D}}_{12}\\ {\stackrel{\_}{C}}_2& {\stackrel{\_.Math.}{D}}_{21}& {\stackrel{\_.Math.}{D}}_{22}\end{array}\right]=\left[\begin{array}{ccc}{A}_0& B_{01}& B_{02}\\ C_{01}& D_{01}& D_{02}\\ C_{02}& D_{03}& D_{04}\end{array}\right]+\left[\begin{array}{c}{T}_1\\ T_5\\ T_6\end{array}\right].Math.\Gamma \left[\begin{array}{ccc}{T}_2& T_3& T_4\end{array}\right].Math..Math..Math.A_0=\left[\begin{array}{cc}A& 0\\ 0& 0\end{array}\right],B_{01}=\left[\begin{array}{cc}0& B_{f.Math..Math.1}\\ 0& 0\end{array}\right],B_{02}=\left[\begin{array}{c}{B}_{f.Math..Math.2}\\ 0\end{array}\right]& \left(10\right)\\ .Math.C_{01}=\left[\begin{array}{cc}0& 0\\ C_{f.Math..Math.1}& 0\end{array}\right],D_{01}=\left[\begin{array}{cc}0& 0\\ 0& D_{f.Math..Math.11}\end{array}\right],D_{02}=\left[\begin{array}{c}0\\ D_{f.Math..Math.12}\end{array}\right].Math..Math..Math.C_{02}=\left[\begin{array}{cc}0& 0\end{array}\right],D_{03}=\left[\begin{array}{cc}0& 0\end{array}\right],D_{04}=D_{22}& \left(11\right)\\ .Math.T_1=\left[\begin{array}{ccc}0& 0& 0\\ I& 0& 0\end{array}\right],T_2=\left[\begin{array}{cc}0& I\\ C_3& 0\\ 0& 0\end{array}\right],T_3=\left[\begin{array}{cc}0& 0\\ 0& D_{31}\\ I& 0\end{array}\right].Math..Math..Math.T_4=\left[\begin{array}{c}0\\ D_{32}\\ 0\end{array}\right],T_5=\left[\begin{array}{ccc}0& 0& I\\ 0& D_{13}& 0\end{array}\right],T_6=\left[\begin{array}{ccc}0& D_{23}& 0\end{array}\right]& \left(12\right)\\ .Math.\Gamma =\left[\begin{array}{ccc}{A}_e& B_{e.Math..Math.1}& B_{e.Math..Math.2}\\ C_{e.Math..Math.1}& D_{e.Math..Math.11}& D_{e.Math..Math.12}\\ C_{e.Math..Math.2}& D_{e.Math..Math.21}& D_{e.Math..Math.22}\end{array}\right].Math..Math.C_3=\left[\begin{array}{c}0\\ C_{f.Math..Math.2}\end{array}\right],D_{22}=\left[\begin{array}{cc}0& -I\\ 0& 0\end{array}\right],.Math..Math.D_{23}=\left[\begin{array}{c}I\\ 0\end{array}\right],D_{31}=\left[\begin{array}{c}0\\ D_{f.Math..Math.21}\end{array}\right],D_{32}=\left[\begin{array}{ccc}I& 0& 0\\ & D_{f.Math..Math.22}& \end{array}\right]& \left(13\right)\end{array}$

and Γ are estimation matrices of the fault estimator.

[0054] Setting

[00031]$X=\left[\begin{array}{cc}L& V\\ V^T& Y\end{array}\right],X^{-1}=\left[\begin{array}{cc}J& W\\ W^T& Z\end{array}\right]$

wherein L, V and Y respectively represent sub-block matrices of X; and J, W and Z respectively represent sub-block matrices of X.

[0055] A matrix P and its inverse matrix {tilde over (P)} are constructed:

[00032]$P=\left[\begin{array}{cc}Q& S\\ S^T& R\end{array}\right]=\left[\begin{array}{cccc}{Q}_1& Q_2& S_1& S_2\\ Q_2^T& Q_3& S_3& S_4\\ S_1^T& S_3^T& R_1& R_2\\ S_2^T& S_4^T& R_2^T& R_3\end{array}\right],.Math.P^{-1}=\tilde{P}=\left[\begin{array}{cc}\tilde{Q}& \tilde{S}\\ {\tilde{S}}^T& \tilde{R}\end{array}\right]=\left[\begin{array}{cccc}{\tilde{Q}}_1& {\tilde{Q}}_2& {\tilde{S}}_1& {\tilde{S}}_2\\ {\tilde{Q}}_2^T& {\tilde{Q}}_3& {\tilde{S}}_3& {\tilde{S}}_4\\ {\tilde{S}}_1^T& {\tilde{S}}_3^T& {\tilde{R}}_1& {\tilde{R}}_2\\ {\tilde{S}}_2^T& {\tilde{S}}_4^T& {\tilde{R}}_2^T& {\tilde{R}}_3\end{array}\right]$

wherein Q.sub.1, Q.sub.2 and Q.sub.3 respectively represent sub-block matrices of Q; S.sub.1, S.sub.2, S.sub.3 and S.sub.4 respectively represent sub-block matrices of S; R.sub.1, R.sub.2 and R.sub.3 respectively represent sub-block matrices of R; {tilde over (Q)}, {tilde over (S)} and {tilde over (R)} respectively represent sub-block matrices of {tilde over (P)}; {tilde over (Q)}.sub.1, {tilde over (Q)}.sub.2 and {tilde over (Q)}.sub.3 respectively represent sub-block matrices of {tilde over (Q)}; {tilde over (S)}.sub.1, {tilde over (S)}.sub.2, {tilde over (S)}.sub.3 and {tilde over (S)}.sub.4 respectively represent sub-block matrices of {tilde over (S)}; {tilde over (R)}.sub.1, {tilde over (R)}.sub.2 and {tilde over (R)}.sub.3 respectively represent sub-block matrices of {tilde over (R)}.

[0056] The following LMIs are constructed, and combined to solve corresponding matrix solutions L, J, Q.sub.3, R.sub.3, S.sub.4, {tilde over (Q)}.sub.3, {tilde over (R)}.sub.3 and {tilde over (S)}.sub.4:

[00033]${N_L^T\left[\begin{array}{ccc}I& 0& 0\\ 0& I& 0\\ 0& 0& I\\ A& B_{f.Math..Math.1}& B_{f.Math..Math.2}\\ C_{f.Math..Math.1}& D_{f.Math..Math.11}& D_{f.Math..Math.12}\\ 0& 0& D_{22}\end{array}\right]}^T\left[\begin{array}{cccccc}0& 0& 0& L& 0& 0\\ 0& Q_3& 0& 0& S_4& 0\\ 0& 0& -\gamma .Math..Math.I& 0& 0& 0\\ L& 0& 0& 0& 0& 0\\ 0& S_4^T& 0& 0& R_3& 0\\ 0& 0& 0& 0& 0& {\gamma }^{-1}.Math.I\end{array}\right].Math.\hspace{1em}\left[\begin{array}{ccc}I& 0& 0\\ 0& I& 0\\ 0& 0& I\\ A& B_{f.Math..Math.1}& B_{f.Math..Math.2}\\ C_{f.Math..Math.1}& D_{f.Math..Math.11}& D_{f.Math..Math.12}\\ 0& 0& D_{22}\end{array}\right].Math.N_L<0.Math..Math.{N_J^T\left[\begin{array}{ccc}-A^T& -C_{f.Math..Math.1}^T& 0\\ -B_{f.Math..Math.1}^T& -D_{f.Math..Math.11}^T& 0\\ -B_{f.Math..Math.2}^T& -D_{f.Math..Math.12}^T& -D_{22}^T\\ I& 0& 0\\ 0& I& 0\\ 0& 0& I\end{array}\right]}^T\left[\begin{array}{cccccc}0& 0& 0& J& 0& 0\\ 0& {\tilde{Q}}_3& 0& 0& {\tilde{S}}_4& 0\\ 0& 0& -{\gamma }^{-1}.Math..Math.I& 0& 0& 0\\ J& 0& 0& 0& 0& 0\\ 0& {\tilde{S}}_4^T& 0& 0& {\tilde{R}}_3& 0\\ 0& 0& 0& 0& 0& \gamma .Math..Math.I\end{array}\right].Math.\hspace{1em}\left[\begin{array}{ccc}-A^T& -C_{f.Math..Math.1}^T& 0\\ -B_{f.Math..Math.1}^T& -D_{f.Math..Math.11}^T& 0\\ -B_{f.Math..Math.2}^T& -D_{f.Math..Math.12}^T& -D_{22}^T\\ I& 0& 0\\ 0& I& 0\\ 0& 0& I\end{array}\right].Math.N_L<0.Math..Math.\left[\begin{array}{cc}J& I\\ I& L\end{array}\right]>0,R>0,Q=-R,S+S^T=0$