Newton-based extremum seeking for higher derivatives of unknown Maps with Delays

Adaptive Control & Signal

Feiling

Koga

et al. 2020

Self Cite

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.

Section: G(t) = M(t)y(t) ĥ(T) = N(t)y(t) (100)mentioning

confidence: 83%

Section: Introductionmentioning

confidence: 83%

“…The principal change is the redefinition of the signals M ( t ), N ( t ),

Ĥ (t)

, G ( t ), and z ( t ). In this sense, we first recall the following auxiliary signal from Reference 23:

\begin{align} {normalΥ}_{j} false(t false) = C_{j} normalsin ((), j ω t + \frac{π}{4} {false(prefix- 1 false)}^{n + 1} ((), 1 + {false(prefix- 1 false)}^{j})), \end{align}

\begin{align} C_{j} = \frac{2^{j} j!}{a^{j}} {false(prefix- 1 false)}^{F}, F = \frac{j prefix- (||, normalsin ((), \frac{j π}{2}))}{2} . \end{align}

In References 12 and 23, it was defined the signal

\hat{h^{(j)}} (t) : = y (t) Υ_{j} (t)

\begin{align} M false(t false) = {normalΥ}_{n + 1} false(t false), N false(t false) = {normalΥ}_{n + 2} false(t false), \end{align}

and, consequently, the gradient and Hessian estimates for Q ( n ...…”

Section: Extension To Newton‐based Es For Maximizing Higher Derivatives Of Unknown Maps With Parabolic Pdesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Adaptive Control & Signal

Feiling

Koga

et al. 2020

Self Cite

“…Embora a busca extremal tenha sido empregada com sucesso em muitas aplicações de engenharia, os autores em (Oliveira et al (2017), Rusiti et al (2018) e Rusiti et al (2019)) apontam a presença de atraso como um fator limitante na aplicação do método em situações práticas. Estas publicações proveram uma análise detalhada e completa da aplicação de projetos para compensação de atrasos em busca extremal com mapeamentos estáticos e dinâmicos.…”

Section: Introductionunclassified

Compensação de EDPs da onda na dinâmica de atuação para busca extremal

Anais Do 14º Simpósio Brasileiro De Automação Inteligente

Albuquerque

2019

Gradient extremum seeking for compensating wave actuator dynamics in cascade with static scalar maps is addressed in the present paper. This class of Partial Differential Equations (PDEs) for extremum seeking has not been studied yet. A dynamic feedback control law based on distributed parameters is proposed by employing backstepping transformation with an appropriate target system and an adequate formulation using Neumann interconnections. Local stability and convergence to a small neighborhood of the desired (but unknown) extremum is proved by means of a Lyapunov functional and the theory of averaging in infinite dimensions. Numerical simulations illustrate the theoretical results. Resumo: Neste artigo é apresentado um controlador por busca extremal baseado no método gradiente para compensar a dinâmica de atuação com dimensão infinita do tipo equação da onda em cascata com um mapeamento escalar estático desconhecido. Esta classe de Equações Diferenciais Parciais (EDPs) ainda não havia sido estudada para busca extremal. Uma lei de controle dinâmica baseada em parâmetros distribuídos é proposta fazendo uso da transformação backstepping com sistema alvo apropriado e adequada formulação usando interconexões de Neumann. A estabilidade local e a convergência para uma pequena vizinhança do extremo (desconhecido) desejado são provadas por meio de um funcional de Lyapunov e a teoria da média em dimensões infinitas. Simulações numéricas ilustram os resultados teóricos.

Newton‐based extremum seeking of higher‐derivative maps with time‐varying delays

Rušiti

Adaptive Control & Signal

Krstić

et al. 2020

Self Cite

Summary We present a Newton‐based extremum seeking algorithm for maximizing higher derivatives of unknown maps in the presence of time‐varying delays. Dealing with time‐varying delays has impact in the predictor design in terms of the transport PDE with variable convection speed functions, the backstepping transformation as well as the conditions imposed on the delay. First, the delay can grow at a rate strictly smaller than one but not indefinitely, the delay must remain uniformly bounded. Second, the delay may decrease at any uniformly bounded rate but not indefinitely, that is, it must remain positive. We incorporate a filtered predictor feedback with a perturbation‐based estimate for the Hessian's inverse using a differential Riccati equation, where the convergence rate of the real‐time optimizer can be made user‐assignable, rather than being dependent on the unknown Hessian of the higher‐derivative map. Furthermore, exponential stability and convergence to a small neighborhood of the unknown extremum point are achieved for locally quadratic derivatives by using backstepping transformation and averaging theory in infinite dimensions.