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Filterless Fixed-time Extremum Seeking for Scalar Quadratic Maps

Michael Tang, Department of Electrical and Computer Engineering

Jorge Poveda, Department of Electrical and Computer Engineering

Miroslav Krstic, Department of Mechanical and Aerospace Engineering

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Advisors

Jorge Poveda

UCSD, ECE

Miroslav Krstic

UCSD, MAE

Research supported by NSF Grant CMMI No. 2228791

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Motivation

System

Output

How can the input be designed to optimize the output?

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Extremum Seeking

Output

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Extremum Seeking

Taylor Series:

Average System:

Gradient Flow

Perturbation

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Extremum Seeking

Convergence is exponential at best

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Finite-time Convergence

asymptotic

exponential

Finite time

Finite time: Exact convergence in finite time!

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Fixed-time Convergence

Finite time

Fixed time

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Fixed-time Stability

DEFINITION: The origin is said to be globally finite time stable if it is globally asymptotically stable and there exists a function

such that

If

is uniformly bounded, the origin is globally fixed

time stable (FxTS).

LYAPUNOV CONDITION:

with:

A. Polyakov, "Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems," in IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 2106-2110, Aug. 2012

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Fixed-time Extremum Seeking

Uses filters to approximate the gradient

Fastest

Oscillators

Fast

Low Pass Filter

Normal

FxT Gradient Flow

Semi-global practical FxT convergence to optimizers

Possible without filters?

J. I. Poveda and M. Krstic, "Non-Smooth Extremum Seeking Control with User-Prescribed Convergence", IEEE Transactions on Automatic Control, Vol 66, No. 12, pp. 6156-6163, 2021

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Gradient Flow

Standard ES

AVERAGING

Gradient Flow

FxT Gradient Flow

AVERAGING

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Naïve Approach

ASSUMPTION:

What is the average system?

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Case 1

Consider when

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Case 1

Set

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Case 1

The Lyapunov function

for the average system yields:

if:

OR

Independent of

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Case 2

Consider when

The average system satisfies:

Lyapunov function:

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Case 1 + Case 2

Lyapunov function:

Discrepancy!

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Main Result

Cost:

THEOREM:

If

then

dynamics is semi-globally practically FxTS as

for the filterless FxT ESC

J. I. Poveda and N. Li, “Robust hybrid zero-order optimizationalgorithms with acceleration via averaging in time,” Automatica, vol.123, p. 109361, 2021

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Main Result

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Introducing an offset

Cost:

Smoothens the average system

FxT property is lost!

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Future Work

Scalar, nonquadratic cost functions
Multivariable cost functions
Different cost functions (game theoretic setting)
Leverage averaging methods
Combine with results on FxTS in interconnections

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Thank you

University of California, San Diego

Thank you!

Questions: myt001@ucsd.edu