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.

Aiming at the problem that how to design a controller without using model information for an unmanned helicopter (UH), a novel data-driven method based on model free adaptive control (MFAC) is proposed in this article. A new form of dynamic linearization (DL) equation composed of pseudo-partial derivative (PPD) and external disturbance is built in order to reduce the influence of disturbance on PPD. Since there are internal and external unknowns included in the DL equation, a nominal DL equation without external disturbance item is introduced to make the estimation of the two items decoupled. The estimation algorithm of PPD is designed based on the nominal DL equation, and a discrete extended state observer is applied specifically into the estimation of external disturbance based on the original DL equation. Besides, with the nominal DL equation, a modified MFAC scheme called PID-like MFAC, which considers the historical input and output (I/O) data is given to make a better performance and a more convenient application compared with the typical MFAC scheme. Furthermore, the stability of the control scheme is proved. Finally, comparative simulations are carried out to verify the effectiveness of the proposed control scheme. An experiment based on UH also verifies its practical performance and the theoretical findings.

In this article, a complete parametric proportional-integral observer (PPIO)-based approach is proposed to improve the performance of gradient estimation for the fast extremum seeking (ES) scheme acting on a Hammerstein plant. Unlike the prevailing gradient estimation approach of the fast ES which uses a Luenberger observer without an explicit way to obtain the observer gains, a systemic complete PPIO is established based on a complete parametric solution to a type of generalized Sylvester matrix equations. The proposed PPIO presents complete parameterization of all the gain matrices as well as the left eigenvectors in terms of some sets of design parameters that represent the degrees of design freedom. Then, the gradient estimator is constructed by multiplying the states of PPIO and the demodulation signal. Moreover, a synthetic objective function, which includes weighted performance indices of the transient error and the steady-state accuracy, is formulated. The performance of the gradient estimator is improved by minimizing the synthetic objective function through adjusting the degrees of freedom of the PPIO, and all explicit values of the parametric gain matrices are derived with the adjusted degrees of freedom. In turn, a faster and more accurate gradient estimation scheme can be obtained and significantly improve the convergence of the closed-loop system. Besides, the proposed PPIO-based estimator has excellent performance under the noise condition. Simulation examples and an application to the lean-burn combustion system are used to illustrate the effectiveness of the proposed gradient estimation scheme.

The issue of neural adaptive self-triggered tracking control for uncertain nonlinear systems with input hysteresis is considered. Combining radial basis function neural networks (RBFNNs) and adaptive backstepping technique, an adaptive self-triggered tracking control approach is developed, where the next trigger instant is determined by the current information. Compared with the event-triggered control mechanism, its biggest advantage is that it does not need to continuously monitor the trigger condition of the system, which is convenient for physical realization. By the proposed controller, the hysteresis's effect can be compensated effectively and the tracking error can be bounded by an explicit function of design parameters. Simultaneously, all other signals in the closed-loop system can be remaining bounded. Finally, two examples are presented to verify the effectiveness of the proposed method.

In this article, an adaptive phase compensator and a high-gain optimizer are proposed for the fast extremum-seeking (ES) scheme acting on a Hammerstein plant. Unlike the widely applied three-time scale tuning, a time scale-independent tuning is achieved by restricting the plant to have a Hammerstein structure and elevating the adaptation of the control input which makes it as fast as the dither signal. Albeit the existed fast ES schemes utilize a high-frequency sinusoidal dither in order to achieve fast minimization of the plant’s static nonlinearity, the phase shift of the plant could not be neglected as the frequency is high and the ES may fail to reach the extreme values if the shift is large enough. Thus, an effective adaptive frequency varying phase compensation method is introduced to deal with the phase shift under the assumptions in this article. Moreover, a high-gain optimizer is presented to remove the restriction that the adaptation gain has to be small and could still guarantee the stability properties, which makes the proposed scheme more practical. The effectiveness of the proposed scheme is illustrated by numerical simulations.