Conflict-Averse Multi-Objective Extremum Seeking and Its Application in Wastewater Treatment Processes

This paper addresses multi-objective optimization problems using conflict-averse multi-objective extremum seeking (CAMOES) for unknown static mapping. As for the traditional multi-objective extremum seeking (MOES) methods, the initial conditions affect the optimal solution, which may result in over-optimizing some objectives. To overcome this issue, by minimizing the average loss function with the 2-norm of the worst individual objective, conflict-averse gradient estimator is investigated to determine the weighting factors and regularization parameter adaptively, extenuating the adverse impact of the prior parameters and initial conditions on the performance of extremum seeking. To cope with the integral windup problem for constrained inputs, penalty-function anti-windup mechanism is explored with smoothing saturation function for multiple-input multiple-output (MIMO) extremum seeking. The stability of CAMOES is theoretically analyzed. Some bi-objective optimization problems are conducted, including a numerical example and a practical problem for wastewater treatment processes (WWTPs). The results demonstrate that the proposed CAMOES exhibits competitive performance. Note to Practitioners—The initial conditions may lead to different Pareto solutions, which brings adverse impact on the multi-objective optimization performance in WWTPs. How to mitigate the influence of the initial parameters on the optimization performance and deal with the changing operation condition is the key to balancing the trade-off between the effluent quality (EQ) and energy cost (EC). This paper presents a control optimization scheme with intelligent modeling, extremum seeking optimization and the control part. The ErrCor-RBF neural network is built to predict the EC and EQ in WWTPs. For constraint conditions, the penalty-function anti-windup mechanism is exploited to avoid integral windup problems. The optimal set-points are dynamically adjusted by the proposed CAMOES to cope with the varying influent flow rate. The experimental results prove the proposed scheme has better operation and control performance. This paper proposes conflict-averse multi-objective extremum seeking (CAMOES), which is applied to the bi-objective optimization of WWTPs but can also be applied to other systems with multi-objective optimization tasks. In future research, we will address the time delay problem in WWTPs.