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

Junlin Xiong; Wenjie Liu

doi:10.3934/naco.2020005

Article Contents

2020, Volume 10, Issue 3: 331-343. Doi: 10.3934/naco.2020005

This issue Previous Article On a new smoothing technique for non-smooth, non-convex optimization Next Article Sliding mode control for uncertain T-S fuzzy systems with input and state delays

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

Junlin Xiong^, and
Wenjie Liu

Department of Automation, University of Science and Technology of China, 96 Jinzhai Road, Hefei, 230027, China

^* Corresponding author: Junlin Xiong
^* Corresponding author: Junlin Xiong

Received: May 2019

Revised: July 2019

Published: June 2020

This work was supported by National Natural Science Foundation of China under Grant 61773357

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Abstract

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.

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

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

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Figure 2. Trajectories of system states and their estimations

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

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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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