Shames I, Teixeira A M H, Sandberg H, et al. Distributed fault detection for interconnected second-order systems[J]. Automatica, 2011, 47(12): 2757-2764.

文章目錄

• 1 Introduction
• 2 Problem formulation
• 3 Model-based fault detection preliminaries
• 4 FDI for networked systems
• • 4.1 UIO for position distributed control
  • 4.2 UIO for position-velocity distributed control
  • 4.3 Faulty node removal
• 5 Application to practical examples
• • 5.1 FDI in power networks
  • 5.2 FDI in formations of mobile agents
  • 5.3 Complexity of the FDI method
• 6 Concluding remarks and future directions

SymbolsDescription

$v_i(t)$	a scalar known external input
${\xi }_i(t),{\zeta }_i(t)$	scalar states
$u_i(t)$	control
$f_{\xi k}(t),f_{\zeta k}(t)$	fault signals

1 Introduction

2 Problem formulation

$$\begin{array}{rl}{\dot{\xi }}_i& ={\zeta }_i(t)\\ {\dot{\zeta }}_i& =u_i(t)+v_i(t)\end{array}\text{\hspace{1em}(1)}$$

the linear control law
$$u_i(t)=?{\kappa }_i{\zeta }_i(t)+\sum _{j\in N_i}{w}_{ij}[({\xi }_j(t)?{\xi }_i(t))+\gamma ({\zeta }_j(t)?{\zeta }_i(t))]\text{\hspace{1em}(2)}$$

發(fā)生故障時，系統(tǒng)的表述為
$$\begin{array}{rl}{\dot{\xi }}_i& ={\zeta }_i(t)+f_{\xi k}(t)\\ {\dot{\zeta }}_i& =u_i(t)+v_i(t)+f_{\zeta k}(t)\end{array}$$

The closed-loop dynamis of the networked system in the presence of faults can be rewritten as
$$\begin{array}{rl}\dot{x}(t)& =Ax(t)+Bv(t)+B_{f}f(t)\\ y(t)& =Cx(t)\end{array}\text{\hspace{1em}(3)}$$

3 Model-based fault detection preliminaries

Consider the linear fault-free system under the influence of an unknown input $d(t)\in {\mathbb{R}}^{m?1}$ described by
$$\begin{array}{rl}\dot{x}(t)& =Ax(t)+Bv(t)+Ed(t)\\ y(t)& =Cx(t)\end{array}\text{\hspace{1em}(5)}$$

The system in presence of faults is given by
$$\begin{array}{rl}\dot{x}(t)& =Ax(t)+Bv(t)+Ed(t)+B_{f}f(t)\\ y(t)& =Cx(t)\end{array}\text{\hspace{1em}(6)}$$

A full-order observer for the fault-free system (5) is described by:
$$\begin{array}{rl}\dot{z}(t)& =Fz(t)+TBv(t)+Ky(t)\\ \hat{x}(t)& =z(t)+Hy(t)\end{array}\text{\hspace{1em}(7)}$$

4 FDI for networked systems

4.1 UIO for position distributed control

Consider the networked system introduced in Section 2 with the following control law
$$m_i{u}_i(t)=?d_i{\zeta }_i(t)+\sum _{j\in N_i}{w}_{ij}({\xi }_j(t)?{\xi }_i(t))\text{\hspace{1em}(14)}$$

Note that if the graph is not connected, the networked system (16) can be decomposed into several decoupled subsystems, each corresponding to a connected subset of the network. The conclusion of Theorem 1 then applies to each subsystem.

4.2 UIO for position-velocity distributed control

Now we consider the existence of UIOs for the distributed control law:
$$u_i(t)=\sum _{j\in N_i}{w}_{ij}[({\xi }_j(t)?{\xi }_i(t))+\gamma ({\zeta }_j(t)?{\zeta }_i(t))]\text{\hspace{1em}(20)}$$

4.3 Faulty node removal

5 Application to practical examples

5.1 FDI in power networks

5.2 FDI in formations of mobile agents

5.3 Complexity of the FDI method

6 Concluding remarks and future directions

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