Distributed Observer for Descriptor Linear System: The Luenberger Observer Method

TL;DR

This work addresses distributed state estimation for descriptor linear systems described by E x dot = A x with outputs y = C x, where impulses complicate observer design. It develops two distributed observer frameworks based on the standard decomposition form and the dynamic decomposition form to achieve asymptotic omniscience while reconstructing impulse phenomena. The SDF based design requires v i 2 = n 2 and a coupling gain bound, whereas the DDF based design relaxes this with different Hurwitz and Lyapunov conditions, offering complementary applicability. Simulations on a hydraulic descriptor system and an electrical network demonstrate that each local observer can track the true state and reconstruct impulses, validating the practicality of the proposed methods.

Abstract

This paper concerns the distributed observer for the descriptor linear system. Unlike centralized descriptor system observers, in the case of distributed observers, each agent either finds it difficult to independently eliminate impulses, or the observer dynamics after eliminating pulses cannot be implemented. To overcome this issue, this paper develops the structure of the distributed observer in two different scenarios, and the observer parameters are presented through a novel design. Moreover, we provide two implementation methods for distributed observer in different scenarios. As a result, each local observer has the ability to reconstruct the states of the underlying system, including its impulse phenomenon. Finally, simulation results verify the validity of our results.

Figures (6)

• Figure 1: Hydraulic system and its dynamics described by descriptor system. Sub-figure 1 and sub-figure 2 are the construction and dynamics of hydraulic, respectively; sub-figure 3 shows the communication topology among three local observers.
• Figure 2: Error dynamics between distributed observer and real states. The first row and the second row shows the dynamics of $h_i$, and $p_i,p_B$, respectively, with $i=1,2,3$. Moreover, sub-figure 1, 4 is from local observer 1; sub-figure 2, 5 is from local observer 2; and sub-figure 3, 6 is from local observer 3.
• Figure 3: Trajectories of the real states and the state estimation of distributed observer. The solid line represents the true state, and the dashed line represents the state estimation of the state observer.
• Figure 4: System formulation of electrical network system. Sub-figure 1 shows the incidence matrix, in which $\mathscr{C}_i$, $\mathscr{R}_i$, $\mathscr{L}_i$ stand for the $i$th capacitor, resistor, and inductor, respectively. $A_{\mathscr{E}}$ and $A_{\mathscr{K}}=col\{A_{\mathscr{K}_3},A_{\mathscr{K}_3},A_{\mathscr{K}_3}\}$ are the sub-matrices of $A_{cir}$, where $\mathscr{K}=\mathscr{E},\mathscr{C},\mathscr{R},\mathscr{L}$. $n_{\mathscr{X}\mathcal{Y}}$ represents the node connecting elements $\mathscr{X}$ and $\mathscr{Y}$.
• Figure 5: Trajectories of the states of electrical network system.

Theorems & Definitions (14)

• Definition 1: duan2010analysis
• Definition 2: duan2010analysis
• Definition 3: duan2010analysis
• Lemma 1: duan2010analysis
• Lemma 2: duan2010analysis
• Lemma 3: Han2017A
• Lemma 4
• Theorem 1
• Remark 1
• Lemma 5