Stochastic Extremum Seeking for Dynamic Maps With Delays

2020

IEEE Access

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This paper proposes a gradient extremum seeking method to address locally quadratic static maps in the presence of time-varying delays. Accommodating nonconstant delays has a strong impact in the predictor construction in terms of the associated transport partial differential equation (PDE) with variable convection speeds beyond the restrictions imposed on the delay regarding its arbitrary duration but bounded variation. A novel predictor design using perturbation-based estimates of the unknown Gradient and Hessian of the map must be introduced to handle this variable nature of the delays, which can arise both in the input and output channels of the nonlinear map to be optimized. Local exponential stability and convergence to a small neighborhood of the unknown extremum point are guaranteed. This technical result is assured by using backstepping transformation and averaging theory in infinite dimensions. Implementation aspects of the presented predictor for variable delays as well as the extension to state-dependent delays are also discussed. At last, we introduce the first results in the topic of extremum seeking control for cascades of transport PDEs that are interconnected through boundary conditions. Such a PDE-PDE cascade is useful to represent simultaneous time-and state-dependent delays. A simulation example illustrates the effectiveness of the proposed predictor-based extremum seeking approach for time-delay compensation. INDEX TERMS Extremum seeking, time-varying delays, state-dependent delays, predictor feedback, partial differential equations, averaging in infinite dimensions, backstepping transformations.

“…(21) Hence, from (20) and (21), the average feedback law (19) can be obtained indeed as desired. Consequently,…”

Section: Predictor Feedback With Hessian's Estimatementioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Gradient Extremum Seeking With Nonconstant Delays

Santos

2020

IEEE Access

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“…Proposition 3. The original closed-loop system (10)- (13) with the controller (57) can be written in the forṁ…”

Section: Invoking Averaging Theoremmentioning

confidence: 99%

Extremum seeking for unknown scalar maps in cascade with a class of parabolic partial differential equations

Adaptive Control & Signal

Feiling

Koga

et al. 2020

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Summary We present a generalization of the scalar gradient extremum seeking (ES) algorithm, which maximizes static maps in the presence of infinite‐dimensional dynamics described by parabolic partial differential equations (PDEs). The PDE dynamics contains reaction‐advection‐diffusion (RAD) like terms. Basically, the effects of the PDE dynamics in the additive dither signals are canceled out using the trajectory generation paradigm. Moreover, the inclusion of a boundary control for the PDE process stabilizes the closed‐loop feedback system. By properly demodulating the map output corresponding to the manner in which it is perturbed, the ES algorithm maximizes the output of the unknown map. In particular, our parabolic PDE compensator employs the same perturbation‐based (averaging‐based) estimate for the Hessian of the function to be maximized applied in the previous publications free of PDEs. We prove local stability of the algorithm, real‐time maximization of the map and convergence to a small neighborhood of the desired (unknown) extremum by means of backstepping transformation, Lyapunov functional and the theory of averaging in infinite dimensions. Finally, we present the generalization to the scalar Newton‐based ES algorithm, which maximizes the map's higher derivatives in the presence of RAD‐like dynamics. By modifying the demodulating signals, the ES algorithm maximizes the nth derivative only through measurements of the own map. The Newton‐based ES approach removes the dependence of the convergence rate on the unknown Hessian of the higher derivative, an effort to improve performance and remove limitations of standard gradient‐based ES. Numerical examples support the theoretical results.

“…Embora para a busca extremal com atrasos de entrada e/ou saída muitos projetos de controle baseados em preditores tenham sido desenvolvidos (Rusiti et al, 2018), (Rusiti et al, 2019), só depois que os resultados em (Oliveira et al, 2017) foram publicados soluções rigorosas para sistemas descritos por Equações Diferenciais Parciais (EDPs) surgiram. Em particular, apenas EDPs de transporte hiperbólico de primeira ordem foram originalmente consideradas em (Oliveira et al, 2017) para representar atrasos puros.…”

Section: Introductionunclassified

Busca Extremal para Maximização de Derivadas de Ordem Superior de Mapas Desconhecidos em Cascata com EDPs de Reação-Advecção-Difusão

Anais Do 14º Simpósio Brasileiro De Automação Inteligente

Feiling

Krstić

2019

Self Cite

We present a generalization of the scalar Newton-based extremum seeking algorithm, which maximizes the map's higher derivatives in the presence of Partial Differential Equation (PDE) dynamics described by Reaction-Advection-Diffusion (RAD) equations. Basically, the effects of the PDE dynamics in the additive dither signals are canceled out using the trajectory generation paradigm. Moreover, the inclusion of a boundary control for the RAD process stabilizes the closed-loop feedback system. By properly demodulating the map output corresponding to the manner in which it is perturbed, the extremum seeking algorithm maximizes the n-th derivative only through measurements of the own map. The RAD compensator employs the same perturbation-based (averaging-based) estimate for the Hessian's inverse of the function to be maximized provided by a differential Riccati equation applied in the previous publications free of PDEs. We prove local stability of the algorithm, maximization of the map sensitivity and convergence to a small neighborhood of the desired (unknown) extremum by means of backstepping transformation, Lyapunov functional and the theory of averaging in infinite dimensions. Resumo: Apresenta-se uma generalização para a busca extremal escalar baseada no método de Newton, que maximiza as derivadas do mapa na presença de dinâmica descrita por Equação Diferencial Parcial (EDP) através de equações de Reação-Difusão-Advecção (RDA). Basicamente, o efeito da EDP nos sinais de excitação aditivaé cancelado usando o paradigma de geração de trajetória. Além disso, a inclusão de um controle de fronteira para o processo RDA estabiliza o sistema em malha fechada. O algoritmo de busca extremal maximiza apenas a derivada n-ésima através de medições do próprio mapa através de uma demodulação adequada da saída. O compensador RDA emprega a mesma estimativa baseada em perturbações (baseada na média) para a inversa da Hessiana a ser maximizada por meio de uma equação diferencial de Riccati, já conhecida de publicações anteriores para o cenário livres de EDPs. Ao final, provase a estabilidade local do algoritmo, a maximização da sensibilidade do mapa e convergência para uma pequena vizinhança do extremo desejado (desconhecido) por meio da transformação backstepping, funcional de Lyapunov e a teoria da média em dimensões infinitas.