Distributed Unknown Input Observer Design: A Geometric Approach

Abstract

We present a geometric approach to designing distributed unknown input observers (DUIOs) for linear time-invariant systems, where measurements are distributed across nodes and each node is influenced by \emph{unknown inputs} through distinct channels. The proposed distributed estimation scheme consists of a network of observers, each tasked with reconstructing the entire system state despite having access only to local input-output signals that are individually insufficient for full state observation. Unlike existing methods that impose stringent rank conditions on the input and output matrices at each node, our approach leverages the $(C,A)$-invariant (conditioned invariant) subspace at each node from a geometric perspective. This enables the design of DUIOs in both continuous- and discrete-time settings under relaxed conditions, for which we establish sufficiency and necessity. The effectiveness of our methodology is demonstrated through extensive simulations, including a practical case study on a power grid system.

Figures (10)

• Figure 1: Example of a distributed unknown input observer consisting of 5 sensor nodes and thus a group of 5 local observers $\mathcal{O}_1,\dots \mathcal{O}_5$. The communication graph is represented by the dashed lines.
• Figure 2: Commutative diagram of the restriction and induced map of $A_{L_i}|\mathscr{X}/\mathscr{W}_{g,i}^*$. $\bar{S}_{g,i}$ is defined as the insertion map $\bar{S}_{g,i}:\mathscr{S}_i^*/\mathscr{W}_{g,i}^*\rightarrow\mathscr{X}/\mathscr{W}_{g,i}^*$, where $\mathscr{S}_i^*/\mathscr{W}_{g,i}^*\simeq\frac{\mathscr{X}/\mathscr{W}_{g,i}^*}{\mathscr{X}/\mathscr{S}_i^*}$. $P_{W_{g,i}^*S_i^*}$ is defined as the canonical projection $P_{W_{g,i}^*S_i^*}:\mathscr{X}/\mathscr{W}_{g,i}^*\rightarrow\mathscr{X}/\mathscr{S}_{i}^*$ such that this diagram commutes.
• Figure 3: Lattice diagrams: Construction of the subspace $\mathscr{W}_{g,i}^*$
• Figure 4: Commutative diagram of $\mathscr{W}_{g,i}^*$ decomposition at node $i$.
• Figure 5: Communication topology for numerical example in Section \ref{['sec:numerical_sim']}.

Theorems & Definitions (25)

• Lemma 1
• Lemma 2
• Remark 1
• Theorem 1
• proof
• Remark 2
• Remark 3
• Lemma 3
• Theorem 2
• proof