Extremum-seeking control: convergence improvements and asymptotic stability
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Extremum-seeking control is an adaptive-control methodology that optimizes the steady-state performance of dynamical plants by automated tuning of plant parameters based on measurements. The main advantage of extremum-seeking control compared to many other optimization techniques is that no plant model, or just a relatively simple plant model, is used. This makes extremum-seeking control suitable to optimize the performance of complex systems, for which an accurate model is unavailable, and systems that are subject to unknown disturbances. Due the low requirements about the knowledge of the plant, extremumseeking control can be applied to many engineering domains. Because the vast majority of the performance-related information about the plant is obtained by measurement, the optimization speed of extremum-seeking control is generally lower than the optimization speed of model-based methods. For model-based methods, the information about the plant's dynamic and steady-state behavior is contained in the plant model and is therefore readily available. In this work, we study extremum-seeking control methods that do not require a plant model. These methods are often referred to as black-box methods. We mainly focus on extremum-seeking methods that rely on added perturbations to optimize the steady-state performance of a plant. For certain classes of plants, large-amplitude high-frequency perturbations can be applied to speed up the convergence of the optimization process. However, large-amplitude high-frequency perturbations may be undesirable or inadmissible in practice due to actuator limitations, a high control e ort, and an increased wear of components. Therefore, we aim to enhance the convergence rate of black-box extremum-seeking methods that use small-amplitude low-frequency perturbations. Extremum-seeking control aims to nd the extremum (that is, the minimum or maximum) of the objective function that represents the steady-state relation between the plant parameters and the plant performance, where the extremum correspond to the optimal steady-state performance. Classical perturbationbased extremum-seeking control methods rely on added perturbations to the plant-parameter values to estimate the gradient of the objective function by correlating the perturbations and the corresponding response in the plant-performance signal. This gradient estimate is subsequently used to steer the plant parameters to the extremum of the objective function using a gradient-descent or gradientascent approach. Hence, the obtained convergence rate is dependent on the accuracy of the gradient estimate. As classic methods use the perturbations of the plant-parameter signals to estimate the gradient of the objective function, an accurate gradient estimate is obtained if the perturbation-related content in the plant-parameter signals is high. We point out in this work that, for small-amplitude low-frequency perturbations, the perturbation-related content in the plant-parameter signals is low and a more accurate gradient estimate and a faster convergence may be achieved by using the entire plant-parameter signals (and not only the perturbation signals) to estimate the gradient of the objective function. This is con rmed by simulation. Moreover, the gradient estimate may be further enhanced by the use of curvature information of the objective function, if available. A continuous-time extremum-seeking controller is presented that uses the entire plant-parameter signals to estimate the gradient of the objective function and allows us to incorporate curvature information of the objective function. In addition, an equivalent discrete-time extremum-seeking controller is presented to optimize the steady-state plant performance in a sampled-data setting. Perturbations are used to provide su cient excitation to estimate the gradient (and sometimes higher-order derivatives) of the objective function. One of the drawbacks of added perturbations is that the plant parameters do not converge to their performance-optimizing values. Instead, they converge to a region of the optimum. Perturbations can be omitted for certain classes of plants. Extremum-seeking control methods that rely on the plant-parameter signals to provide su cient excitation without any form of added excitation are referred to as self driving. Although there exist examples in the literature for which selfdriving extremum-seeking control is applied to achieve an optimal steady-state performance, so far, no conditions have been stated under which convergence to the true optimum can be guaranteed. Here, we prove that there exist conditions on the plant and the self-driving extremum-seeking controller under which the plant parameters are certain to converge to their performance-optimizing values. Self-driving extremum-seeking control has its limitations in terms of applicability. Instead of omitting the perturbations, one may gradually reduce the level of the perturbations to zero as time goes to in nity to arrive at the optimal steady-state performance. Several methods to regulate the amplitude of the added perturbations have been proposed in the literature to obtain asymptotic convergence to the optimum. Commonly local convergence is proved, often for a limited class of plants. In this work, we prove that global asymptotic convergence of the plant parameters to their performance-optimizing values can be guaranteed for general nonlinear plants under certain assumptions. The key to this result is that not only the amplitude but also the frequencies of the perturbations, as well as other tuning parameters of the controller, decay to zero as time goes to in nity. Remarkably, the time-varying tuning parameters can be chosen such that global asymptotic convergence is achieved for all plants that satisfy the assumptions, thereby guaranteeing stability of the resulting closed-loop system of plant and controller regardless of tuning. In a case study, we show that extremum-seeking control can be applied to optimize the injection current of an active power lter for system-wide harmonic mitigation in electrical grids. The used extremum-seeking control method can be parallelized under certain design assumptions in order to increase the convergence speed of the method. A case study of a two-bus electrical grid with distributed generators displays an improved performance of the used extremumseeking control method compared to a local- ltering approach under constant load conditions of the electrical grid, while the performance with respect to a model-based system-wide ltering method is comparable. The case study also shows that the used extremum-seeking control method is slower to respond to changes in load conditions than the local and the model-based system-wide ltering methods. The extremum-seeking control method can be implemented on top of existing approaches to combine the fast transient response of conventional harmonic-mitigation methods with the optimizing capabilities of extremumseeking control.