﻿ 广义系统的有限频域故障估计器设计
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 自动化学报  2018, Vol. 44 Issue (3): 545-551 PDF

1. 哈尔滨工业大学航天学院 哈尔滨 150001

Fault Estimator Design for Descriptor Systems in Finite Frequency Domain
WANG Zhen-Hua1, SHEN Yi1
1. School of Astronautics, Harbin Institute of Technology, Harbin 150001
Manuscript received : July 5, 2016, accepted: December 27, 2016.
Foundation Item: Supported by National Natural Science Foundation of China (61403104, 61773145) and the Fundamental Research Funds for the Central Universities (HIT.KLOF.2015.076)
Corresponding author. SHEN Yi   Professor at the School of Astronautics, Harbin Institute of Technology. His research interest covers fault diagnosis, flight vehicle control, and ultrasound signal processing. Corresponding author of this paper
Recommended by Associate Editor JIANG Bin
Abstract: This paper proposes a novel fault estimator design method for linear descriptor systems with actuator fault and unknown disturbance. The proposed fault estimator has a nonsingular structure and hence is easy to implement. Under the condition that faults belong to a finite frequency domain, the generalized Kalman-Yakubovich-Popov (KYP) lemma is used to propose robust design conditions to attenuate the effect of unknown disturbance and faults on the fault estimation error. Moreover, the design conditions are converted to linear matrix inequalities, which can be solved easily. Finally, an electrical circuit is simulated to verify the effectiveness of the proposed method.
Key words: Descriptor systems     fault estimator     generalized Kalman-Yakubovich-Popov (KYP) lemma     finite frequency domain

1 问题描述

 $\begin{cases} E\dot{{\pmb x}}(t) = A{\pmb x}(t) + B{\pmb u}(t) +B_d{\pmb d}(t) +B_f {\pmb f}(t) \\ {\pmb y}(t) = C{\pmb x}(t) + D_{d}{\pmb d}(t) \end{cases}$ (1)

 $TE+NC=I_n$ (8)

 ${\pmb e}(t) = {\pmb x}(t) - \hat{{\pmb x}}(t)$ (9)
 $\tilde{{\pmb f}}(t) = {\pmb f}(t) - \hat{{\pmb f}}(t)$ (10)

 $\dot{{\pmb e}}(t) = \dot{{\pmb x}}(t) - \dot{\hat{{\pmb x}}}(t)= \\ TA{\pmb x}(t) +TB{\pmb u}(t) +TB_d{\pmb d}(t) +TB_f {\pmb f}(t) +\\ NC\dot{{\pmb x}}(t) - TA\hat{{\pmb x}}(t)-TB{\pmb u}(t) -\\ L({\pmb y}(t)-C\hat{{\pmb x}}(t))-N\dot{{\pmb y}}(t) =\\ (TA-LC){\pmb e}(t)+TB_f{\pmb f}(t) + \\ (TB_d -LD_d){\pmb d}(t) -ND_d \dot{{\pmb d}}(t)$ (11)

 $\tilde{{\pmb f}}(t) = {\pmb f}(t) -VC{\pmb e}(t)-VD_d{\pmb d}(t)$ (12)

 $\begin{cases} \dot{{\pmb e}}(t) = (TA-LC){\pmb e}(t)+TB_f{\pmb f}(t) +\\ \qquad\ \ (TB_d-LD_d) {\pmb d}(t) -ND_d \dot{{\pmb d}}(t)\\ \tilde{{\pmb f}}(t) = {\pmb f}(t) - VC{\pmb e}(t)-VD_d{\pmb d}(t) \end{cases}$ (13)

$\bar{{\pmb d}}(t) = \bigl[\begin{matrix} {\pmb d}^{\rm T}(t)&\dot{{\pmb d}}^{\rm T}(t) \end{matrix} \bigr] ^{\rm T}$, 可将误差系统(13)表示成如下形式

 $\begin{cases} \dot{{\pmb e}}(t) = (TA-LC){\pmb e}(t)+TB_f{\pmb f}(t) +\tilde{B}_d\bar{{\pmb d}}(t) \\ \tilde{{\pmb f}}(t) = -VC{\pmb e}(t) + {\pmb f}(t) +\tilde{D}_d \bar{{\pmb d}}(t) \end{cases}$ (14)

 $\Vert G_{\tilde{{f}}{ f}}(s) \Vert^{[-\varpi, \varpi]}_{\infty} < \gamma_f$ (15)
 $\Vert G_{\tilde{{ f}}\bar{{ d}}}(s) \Vert_{\infty} < \gamma_d$ (16)

1) 下面的有限频域不等式成立

 $\left[\begin{matrix} G({\rm j}\omega) \\ I \end{matrix} \right] ^{\rm T} \Pi \left[\begin{matrix} G({\rm j}\omega) \\ I \end{matrix} \right] <0, \ \forall \omega \in \Omega$ (18)

 $\left[\begin{matrix} \mathcal{A} & \mathcal{ B} \\ I & 0 \end{matrix} \right] ^{\rm T} \Xi \left[\begin{matrix} \mathcal{A} & \mathcal{B} \\ I & 0 \end{matrix} \right] + \left[\begin{matrix} \mathcal{C} & \mathcal{D} \\ 0 & I \end{matrix} \right] ^{\rm T} \Pi \left[\begin{matrix} \mathcal{C} & \mathcal{ D} \\ 0 & I \end{matrix} \right] <0$ (19)

1) 矩阵不等式

 $\mathscr{U} ^{\perp} \mathscr{Q} (\mathscr{U}^{\perp})^{\rm T} <0$

2) 存在一个矩阵$\mathscr{Y} \in{\boldsymbol{\rm{R}}}^{ {b \times a}}$, 使得

 $\mathscr{Q}+ \mathscr{U} \mathscr{Y}+\mathscr{Y}^{\rm T} \mathscr{U}^{\rm T} <0$

 $\begin{cases} \dot{{\pmb x}}(t) = \mathcal{A}{\pmb x}(t)+\mathcal{B}{\pmb w}(t)\\ {\pmb y}(t) = \mathcal{C} {\pmb x}(t) + \mathcal{D} {\pmb w}(t) \end{cases}$ (20)

 $G(s) =\mathcal{C} (sI- \mathcal{A})^{-1}\mathcal{B} + \mathcal{D}$ (21)

 $\left[\begin{matrix} \mathcal{A}^{\rm T}\mathcal{P} + \mathcal{P} \mathcal{A} + \mathcal{C}^{\rm T}\mathcal{C} & \mathcal{P} \mathcal{B} +\mathcal{C}^{\rm T}\mathcal{D} \\ \mathcal{B}^{\rm T} \mathcal{P} +\mathcal{D}^{\rm T}\mathcal{C } & \mathcal{D}^{\rm T}\mathcal{D}-\gamma^2I \end{matrix} \right] <0$ (22)

1) $G(s)$满足如下有限频域不等式

 $\left[\begin{matrix} G({\rm j}\omega) \\ I \end{matrix} \right] ^{\rm T} \Pi \left[\begin{matrix} G({\rm j}\omega) \\ I \end{matrix} \right] <0, \ \ \forall \vert \omega \vert \leq \varpi_l$ (23)

 $\left[\begin{matrix} \Theta_l + \bar{\mathcal{M}}\bar{\mathcal{A}} +\bar{\mathcal{A}}^{\rm T}\bar{\mathcal{M}}^{\rm T} & * \\ -\bar{\mathcal{M}}^{\rm T} +\bar{\mathcal{P}}_l^{\rm T} +\mathcal{G} \bar{\mathcal{A}} & -\mathcal{Q}-\mathcal{G}-\mathcal{G}^{\rm T} \end{matrix} \right] <0$ (26)

 $\Theta_m = \left[\begin{matrix} -\varpi_1 \varpi_2 \mathcal{Q} & 0 \\ 0 & 0 \end{matrix} \right] + \left[\begin{matrix} \mathcal{C} & \mathcal{D} \\ 0 & I \end{matrix} \right] ^{\rm T} \Pi \left[\begin{matrix} \mathcal{C} & \mathcal{ D} \\ 0 & I \end{matrix} \right] \\ \bar{\mathcal{P}}_m = \left[\begin{matrix} \mathcal{P}-{\rm j} \varpi _c \mathcal{Q} \\ 0 \end{matrix} \right]$ (27)

3) $G(s)$满足有限频域不等式

 $\left[\begin{matrix} G({\rm j}\omega) \\ I \end{matrix} \right] ^{\rm T} \Pi \left[\begin{matrix} G({\rm j}\omega) \\ I \end{matrix} \right] <0, \ \ \forall \vert \omega \vert \geq \varpi_h$ (29)

 $\Pi =\left[ \begin{matrix} I_q & 0 \\ 0 &-\gamma_f^2 I_q \end{matrix} \right]$

 $\left[\begin{matrix} \varpi^2 \mathcal{Q} +\mathcal{C}^{\rm T}\mathcal{C} +{\rm{He}} \left\{ \mathcal{M}_1\mathcal{A} \right\} & * \\ \mathcal{M}_2\mathcal{A}+ \mathcal{B}^{\rm T}\mathcal{M}^{\rm T}_1 + \mathcal{D}^{\rm T}\mathcal{C} & {\rm{He}}\left\{ \mathcal{M}_2\mathcal{B} \right\} +\mathcal{D}^{\rm T}\mathcal{D}-\gamma_f^2I_q \\ -\mathcal{M}^{\rm T}_1 + \mathcal{P} +\mathcal{G} \mathcal{A} &- \mathcal{M}^{\rm T}_2 +\mathcal{G} \mathcal{B} \end{matrix} \right. \\ \left.\begin{matrix} & * \\ & * \\ & -\mathcal{Q}-\mathcal{G}-\mathcal{G}^{\rm T} \end{matrix} \right] <0$ (39)

 $\mathcal{A} = TA- LC, \ \mathcal{B} = TB_f, \ \mathcal{C} =-VC , \ \mathcal{D} = I_q \\ \mathcal{Q}=Q, \ \mathcal{P} =P_1, \ \mathcal{M}_1 =\alpha_1G, \ \mathcal{M}_2 = V_1^{\rm T} G, \ \mathcal{G}=G$

 $\left[\begin{matrix} \Phi_{11} & * & * \\ \Phi_{21} & \Phi_{22} & * \\ \Phi_{31} & \Phi_{32} & \Phi_{33} \end{matrix} \right] <0$ (40)

 $\Phi_{11} = \varpi^2 Q +(VC)^{\rm T}VC +{\rm{He}}\left\{ \alpha_1 G(T A -LC) \right\} \\ \Phi_{21} = \alpha_1 TB_f^{\rm T}G^{\rm T} + V_1^{\rm T}G(TA-LC) \\ \Phi_{22} = -\gamma_f^2 I_q + V_1^{\rm T}G TB_f + (TB_f)^{\rm T}G^{\rm T} V_1 \\ \Phi_{31} = -\alpha_1G^{\rm T}+ G( TA-LC) +P_{1} \\ \Phi_{32} = -G^{\rm T} V_1+ GTB_f\\ \Phi_{33} = - Q-G -G^{\rm T}$

 $\left[\begin{matrix} \tilde{\Psi}_{11} & * & * & * & * \\ \tilde{\Psi}_{21} &-\gamma_d^2 I_l & * & * & * \\ \tilde{\Psi}_{31} & 0 &-\gamma_d^2 I_l & * & * \\ \tilde{\Psi}_{41} & \tilde{\Psi}_{42} &-GND_d & \tilde{\Psi}_{44} & * \\ -VC & -VD_d & 0 & 0 & -I_q \end{matrix} \right] <0$ (41)

 $\tilde{\Psi}_{11} = {\rm{He}}\left\{ \alpha_2 G(T A-L C) \right\} \\ \tilde{\Psi}_{21} = \alpha_2 (TB_d-LD_d)^{\rm T}G^{\rm T} \\ \tilde{\Psi}_{31} = -\alpha_2 (ND_d)^{\rm T}G^{\rm T} \\ \tilde{\Psi}_{41} = -\alpha_2 G^{\rm T}+ G( TA-LC) + P_{2} \\ \tilde{\Psi}_{42} = G(TB_d-LD_d)\\ \tilde{\Psi}_{44} = -G-G^{\rm T}$

 $\mathcal{Q} = 0, \quad \Pi =\left[ \begin{matrix} I_q & 0 \\ 0 &-\gamma_d^2 I_l \end{matrix} \right]$

 $\left[\begin{matrix} \mathcal{C}^{\rm T}\mathcal{C} +{\rm{He}}\left\{ \mathcal{M}_1\mathcal{A} \right\} & * \\ \mathcal{M}_2\mathcal{A}+ \mathcal{B}^{\rm T}\mathcal{M}^{\rm T}_1 + \mathcal{D}^{\rm T} \mathcal{C} & {\rm{He}}\left\{ \mathcal{M}_2\mathcal{B} \right\} +\mathcal{D}^{\rm T} \mathcal{D}-\gamma_d^2I_l \\ -\mathcal{M}^{\rm T}_1 + \mathcal{P} +\mathcal{G} \mathcal{A} &- \mathcal{M}^{\rm T}_2 +\mathcal{G} \mathcal{B} \end{matrix} \right. \\ \left.\begin{matrix}& * \\& * \\&-\mathcal{G}-\mathcal{G}^{\rm T}\end{matrix} \right] <0$ (42)

 $\mathcal{A} = TA- LC, \ \mathcal{B} = \tilde{B}_d, \ \mathcal{C} =-VC, \ \mathcal{D} =\tilde{D}_d \\ \mathcal{P} =P_2, \ \mathcal{M}_1 =\alpha_2G, \ \mathcal{M}_2 = 0, \ \mathcal{G}=G$

 $\left[\begin{matrix} \tilde{\phi}_{11} & * & * & * \\ \tilde{\phi}_{21} & \tilde{\phi}_{22} & * & * \\ \tilde{\phi}_{31} &\tilde{\phi}_{32} & \tilde{\phi}_{33} & * \\ -VC & I_q & 0 &-I_q \end{matrix} \right] <0$ (47)
 $\left[\begin{matrix} \tilde{\psi}_{11} & * & * & * \\ \tilde{\psi}_{21} &-\gamma_d^2 I_l & * & * \\ \tilde{\psi}_{31} & \tilde{\psi}_{32} &-G-G^{\rm T} & * \\ -VC &-VD_d & 0 & -I_q \end{matrix} \right] <0$ (48)

 $\tilde{\phi}_{11} = \varpi^2 Q +{\rm{He}}\left\{ \alpha_1 (G A-W C) \right\} \\ \tilde{\phi}_{21} = \alpha_1 B_f^{\rm T}G^{\rm T} + V_1^{\rm T} (G A-W C)\\ {\tilde{\phi}_{22}} = -{\gamma_f^2 I_q + V_1^{\rm T}G B_f + B_f^{\rm T}G^{\rm T} V_1} \\ \tilde{\phi}_{31} = -\alpha_1G^{\rm T}+ G A-W C + P_{1} \\ {\tilde{\phi}_{32}}= -{G^{\rm T} V_1+ GB_f} \\ {\tilde{\phi}_{33}} = -{Q-G -G^{\rm T}}\\ \tilde{\psi}_{11} = {\rm{He}}\left\{ \alpha_2 (G A-W C) \right\} \\ \tilde{\psi}_{21} = \alpha_2 (GB_d-WD_d)^{\rm T} \\ { \tilde{\psi}_{31} }= -\alpha_2 G^{\rm T}+ GA-W C + P_{2} \\ { \tilde{\psi}_{32} } = GB_d-WD_d$

3 仿真结果

 ${\pmb f}(t) = \begin{cases} 0, & t <31 \mathit{\boldsymbol{s}} \\ -0.4\sin(0.1t-3.1), & t\geq 31 \mathit{\boldsymbol{s}} \end{cases}$

 图 2 实际故障在设计频域范围内时的故障估计结果 Figure 2 Fault estimation results when the fault is actually in the designed frequency range

 ${\pmb f}(t) = \begin{cases} 0, & t <31 \mathit{\boldsymbol{s}} \\ -0.2\sin(t-31), & t\geq 31 \mathit{\boldsymbol{s}} \end{cases}$

 图 3 实际故障频率超出设计频域范围时的故障估计结果 Figure 3 Fault estimation results when the fault is not actually in the designed frequency range
4 结论

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