Distributed Wireless Networked 𝐻_∞ Control for a Class of Lurie-Type Nonlinear Systems

Algorithm 2

Minimizing 𝛾 to solve the optimal 𝐻_∞ WNCN for closed-loop system 𝒢.

Step  1. Set 𝑘=0. Let 𝛾⁰₋ and 𝛾⁰₊ be the initial lower and upper bounds of 𝛾, that is, 𝛾₀∈[𝛾⁰₋,𝛾⁰₊] where 𝛾⁰₋=0, 𝛾⁰₊ can be

   assigned an arbitrarily sufficiently large value to make inequalities (25)-(26) have initial feasible solution set

    Υ₀=𝑃,𝑄,𝑅,Σ,𝑊^𝒪,𝑊^𝒞,𝑊^ℐ.

Step  2. At 𝑘≥0, compute 𝛾_𝑘=(𝛾^𝑘₋+𝛾^𝑘₊)/2.

Step  3. Use Algorithm 1 to check whether the feasible solution set Υ_𝑘+1 satisfying inequality (15). If Υ_𝑘+1 exists, set 𝛾^𝑘₊=𝛾^𝑘.

   Otherwise, set 𝛾^𝑘₋=𝛾^𝑘.

Step  4. If 𝛾^𝑘₊−𝛾^𝑘₋≤𝜖, where 𝜖 is the assumed calculation accuracy, set optimal 𝐻_∞ performance index 𝛾^∗=𝛾^𝑘₊ and exit.

   Otherwise, set 𝑘=𝑘+1, go to Step  2.