Distributed algorithms for aggregative games of multiple heterogeneous Euler–Lagrange systems

Distributed adaptive Nash equilibrium seeking and disturbance rejection for noncooperative games of high‐order nonlinear systems with input saturation and input delay

“…In electricity market, the competition among distributed energy resources can be described by noncooperative games of generation systems. 23 Consider a noncooperative game with six turbine-generator systems, that is, N = 6, described by Deng and Liang 23 where P e i , X e i , and i ∈ R denote the output power, the valve opening and the relative speed of the generation system i, i = 1, … , 6, respectively; K m i , T m i , and R i are the gain, the time constant, and the regulation constant of the ith machine's turbine, respectively; K e i and T e i stand for the gain and the time constant of the speed governor, respectively; D = 5.0 and H = 4.0 denote the damping constant and the inertia constant, respectively; 0 = 314.159 is the synchronous machine speed; and u e i ∈ R is the control input with i = 0.10s being the time delay. The communication topology among the generation systems is shown in Figure 5.…”

“…) 2 is the generation cost and the term 200 − ∑ N j=1 P e j ∕10 denotes the electricity price. 23 It can be seen from (40), the payoff function of generation system i involves all the generation systems' output powers, and thereby the ADO ( 15) is required when solving the game problem (40).…”

“…(1) This article formulates a noncooperative game for high-order nonlinear systems, and thereby the considered problem can cover the existing works associated with first-order systems, [16][17][18][19] second-order systems, 22,23 and multiintegrator systems 24 as special cases. Moreover, the focused players are subject to input saturation, input delay and external disturbances, which renders the existing seeking algorithms not applicable to the considered problem.…”

“…In Reference 22, input saturation was considered for noncooperative games with double‐integrator dynamics. The authors of Reference 23 further investigated the Nash equilibrium seeking problem for aggregative games of Euler–Lagrange systems. In Reference 24, Nash equilibrium seeking algorithms were developed based on the gradient play technique for games played by multiintegrator agents subject to external disturbances.…”

“…In Reference 24, Nash equilibrium seeking algorithms were developed based on the gradient play technique for games played by multiintegrator agents subject to external disturbances. The works in References 22‐24 are applicable to address the game problems with second‐ or high‐order dynamics but still have the following limitations. Nonlinear dynamics are not considered for the focused MASs. The developed seeking algorithms require full‐state measurements that are not available in many practical applications, especially for high‐order systems. The design of the seeking algorithms depends on the knowledge of the Laplacian matrix associated with the communication topology.…”