Exponential and Prescribed-Time Extremum Seeking with Unbiased Convergence

Abstract

We present multivariable extremum seeking (ES) designs that achieve unbiased convergence to the optimum. Two designs are introduced: one with exponential unbiased convergence (unbiased extremum seeker, uES) and the other with user-assignable prescribed-time unbiased convergence (unbiased PT extremum seeker, uPT-ES). In contrast to the conventional ES, which uses persistent sinusoids and results in steady-state oscillations around the optimum, the exponential uES employs an exponentially decaying amplitude in the perturbation signal (for achieving convergence) and an exponentially growing demodulation signal (for making the convergence unbiased). The achievement of unbiased convergence also entails employing an adaptation gain that is sufficiently large in relation to the decay rate of the perturbation amplitude. Stated concisely, the bias is eliminated by having the learning process outpace the waning of the perturbation. The other algorithm, uPT-ES, employs prescribed-time convergent/blow-up functions in place of constant amplitudes of sinusoids, and it also replaces constant-frequency sinusoids with chirp signals whose frequency grows over time. Among the convergence results in the ES literature, uPT-ES may be the strongest yet in terms of the convergence rate (prescribed-time) and accuracy (unbiased). To enhance the robustness of uES to a time-varying optimum, exponential functions are modified to keep oscillations at steady state. Stability analysis of the designs is based on a state transformation, averaging, local exponential/PT stability of the averaged system, local stability of the transformed system, and local exponential/PT stability of the original system. For numerical implementation of the developed ES schemes and comparison with previous ES designs, the problem of source seeking by a two-dimensional velocity-actuated point mass is considered.

Figures (7)

• Figure 1: Exponential uES scheme. The design uses an exponential decay function $\alpha$ to gradually reduce the effect of the perturbation signal $S(t)$ and its multiplicative inverse $\frac{1}{\alpha}$ to gradually increase the effect of the demodulation signal $M(t)$.
• Figure 2: uPT-ES scheme. The design modifies the exponential uES in Fig. \ref{['ESBlock']} by incorporating $\mu^q$, with $q \geq 1$, into all system dynamics and using hyperbolic chirps in the perturbation and demodulation signals.
• Figure 3: The developed ES scheme for velocity-actuated point mass. For exponential convergence, choose $\mu \equiv 1, \alpha=\alpha_0 e^{-\lambda t}$$\forall t \in [0,\infty)$ and for prescribed-time convergence, choose $\mu=\frac{T}{T-t}, \alpha=\alpha_0 e^{\lambda T \left(1-\frac{T}{T-t}\right)}$$\forall t \in [0,T)$ with $\alpha_0, \lambda>0$.
• Figure 4: Static source seeking by an autonomous vehicle \ref{['velactmass']}. The nominal ES with low amplitude $\alpha_0$ approaches the source more closely but requires high initial velocity, leading to initial deviation from the source. The exponential uES \ref{['ffed']}, with its exponentially decaying amplitude, avoids this issue.
• Figure 5: Time-varying source seeking by an autonomous vehicle \ref{['velactmass']}. The trajectory of the source is illustrated by yellow circles at 10 second intervals. The robust exponential design \ref{['closedrobust']} successfully tracks the source, owing to its amplitude that decays but does not vanish, while exponential design \ref{['ffed']} fails to track.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4
• Theorem 5