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Nash equilibrium seeking with prescribed performance

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Abstract

In this work, we study a Nash equilibrium (NE) seeking problem for strongly monotone non-cooperative games with prescribed performance. Unlike general NE seeking algorithms, the proposed prescribed-performance NE seeking laws ensure that the convergence error evolves within a predefined region. Thus, the settling time, convergence rate, and maximum overshoot of the algorithm can be guaranteed. First, we develop a second-order Newton-like algorithm that can guarantee prescribed performance and asymptotically converge to the NE of the game. Then, we develop a first-order gradient-based algorithm. To remove some restrictions on this first-order algorithm, we propose two discontinuous dynamical system-based algorithms using tools from non-smooth analysis and adaptive control. We study the special case in optimization problems. Then, we investigate the robustness of the algorithms. It can be proven that the proposed algorithms can guarantee asymptotic convergence to the Nash equilibrium with prescribed performance in the presence of bounded disturbances. Furthermore, we consider a second-order dynamical system solution. The simulation results verify the effectiveness and efficiency of the algorithms, in terms of their convergence rate and disturbance rejection ability.

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Notes

  1. In addition to games, optimization problems are also involved in this work. Unless otherwise specified, the statement “specific objectives" refers to the specific objectives for games.

  2. A map \(F: \mathbb {R}^N\rightarrow \mathbb {R}^N\) is monotone if for all \(x,y\in \mathbb {R}^N\), \((F(x)-F(y))^{\textrm{T}}(x-y)\ge 0\). It is strictly monotone if the inequality is strict when \(x\ne y\). F is strongly monotone if there exists a positive constant m such that \((F(x)-F(y))^{\textrm{T}}(x-y)\ge m\Vert x-y\Vert ^2\) for all \(x,y\in \mathbb {R}^N\) [25].

  3. J is invertible based on Assumptions 1–2 and Lemma 2, because \(Jz=0\) if and only if \(z=0\). The algorithm is well defined, and we prove that S is bounded in the proof. In addition, the right-hand side of the system is bounded and will not tend to infinity.

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Authors and Affiliations

  1. Continental-NTU Corporate Lab, Nanyang Technological University, 639798, Nanyang, Singapore

    Chao Sun & Guoqiang Hu

  2. School of Electrical and Electronic Engineering, Nanyang Technological University, 639798, Nanyang, Singapore

    Guoqiang Hu

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  1. Chao Sun
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  2. Guoqiang Hu
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Correspondence to Guoqiang Hu.

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This work was supported by the RIE2020 Industry Alignment Fund—Industry Collaboration Projects (IAF-ICP) Funding Initiative, as well as cash and in-kind contribution from the industry partner(s).

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Sun, C., Hu, G. Nash equilibrium seeking with prescribed performance. Control Theory Technol. 21, 437–447 (2023). https://doi.org/10.1007/s11768-023-00169-4

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