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# $H_{\infty}$ observer-based control for large-scale systems with sparse observer communication network

• * Corresponding author: Junlin Xiong
This work was supported by National Natural Science Foundation of China under Grant 61773357
• This paper studies the $H_{\infty}$ control problem for large-scale systems under a sparse observer communication network. Different from existing approaches, where the topology of the observer communication network is fixed, we aim to design the sparse observer communication network such that the closed-loop system is asymptotically stable and satisfies the $H_{\infty}$ performance. Firstly, sufficient conditions are established to design the distributed $H_{\infty}$ observer and controller gains in terms of LMIs. Then, the developed sufficient conditions are used to minimize the number of the links in the observer communication network. Two numerical algorithms are proposed to solve the sparse observer communication network design problem. Finally, a numerical example is given to demonstrate the effectiveness of the proposed approach.

Mathematics Subject Classification: Primary: 93C15, 93B36; Secondary: 49N35.

 Citation: • • Figure 1.  Sparse observer topology structure of large-scale systems

Figure 2.  Trajectories of system states and their estimations

Figure 3.  Number of observer network links for different $H_{\infty}$ performance levels

 Algorithm 2 Sparse Observer Communication Network Link Design (1) Choose a small enough scalar $\delta>0$. Initialize $\theta_{ij}^{(0)}=1$ for all $i\neq j$. Let $k\leftarrow1$. (2) Solve the optimization problem (12) with $\theta_{ij}=\theta_{ij}^{(k-1)}$ to find the optimal solution $Z^{(k)}$. (3) Construct the index set $I=\left\{{(i,j) \mid \left\lVert{\mathrm{vec}(Z^{(k)}_{ij})}\right\rVert_{1}<\delta, \theta_{ij}=1, i\neq j}\right\}$. If $I$ is an empty set, then go to step (5). Otherwise, select $(i^{*},j^{*}) = {\rm{arg\;min}}_{(i,j)\in I} \left\lVert{\mathrm{vec}(Z^{(k)}_{ij})}\right\rVert_{1}$. (4) Update the links $\theta_{ij}^{(k)}=\theta_{ij}^{(k-1)}$ except that $\theta_{i^{*}j^{*}}^{(k)}=0$. Let $k\leftarrow k+1$ and go to Step (2). (5) Let $\theta_{ij}=\theta_{ij}^{(k-1)}$. Solve the LMI (9) and design the controller and observer gains as $K_{i}=Y_{i}X_{i}^{-1}$, $L_{ij}=Q_{ij}^{-1}Z_{ij}$, respectively.
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