Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Practical prescribed-time prescribed performance control with asymptotic convergence -- A vanishing sigma-modification approach

Mehdi Golestani, Yongduan Song, Weizhen Liu, Guangren Duan, He Kong

TL;DR

The paper addresses adaptive control of uncertain nonlinear systems with time-varying parametric and dynamic uncertainties, seeking guaranteed performance within a prescribed time horizon $T$. It introduces an adaptive dynamic surface control framework that uses a performance-rate function $μ(t)$ that freezes at $T$ and a vanishing leakage term $σ_2(t)$ in a $\sigma$-modification to achieve practical prescribed-time convergence to a small region and eventual asymptotic convergence. Barrier Lyapunov functions and a prescribed-time nonlinear filter are employed to guarantee prescribed performance and cope with nonvanishing unmodeled dynamics that satisfy exp-ISpS. Simulation results validate effectiveness, show reduced initial control effort, and illustrate potential applications in aerospace attitude control and robotic surgery.

Abstract

In this paper, we present a method capable of ensuring practical prescribed-time control with guaranteed performance for a class of nonlinear systems in the presence of time-varying parametric and dynamic uncertainties, and uncertain control coefficients. Our design consists of two key steps. First, we construct a performance-rate function that freezes at and after a user-specified time T, playing a crucial role in achieving desired precision within prescribed time T and dealing with unmodeled dynamics. Next, based on this function and a sigma-modification strategy in which the leakage term starts to vanish at t > T, we develop an adaptive dynamic surface control framework to reduce control complexity, deal with uncertainties, ensure prescribed performance, practical prescribed-time convergence to a specific region, and ultimately achieve asymptotic convergence. The effectiveness of the proposed control method is validated through numerical simulations.

Practical prescribed-time prescribed performance control with asymptotic convergence -- A vanishing sigma-modification approach

TL;DR

The paper addresses adaptive control of uncertain nonlinear systems with time-varying parametric and dynamic uncertainties, seeking guaranteed performance within a prescribed time horizon . It introduces an adaptive dynamic surface control framework that uses a performance-rate function that freezes at and a vanishing leakage term in a -modification to achieve practical prescribed-time convergence to a small region and eventual asymptotic convergence. Barrier Lyapunov functions and a prescribed-time nonlinear filter are employed to guarantee prescribed performance and cope with nonvanishing unmodeled dynamics that satisfy exp-ISpS. Simulation results validate effectiveness, show reduced initial control effort, and illustrate potential applications in aerospace attitude control and robotic surgery.

Abstract

In this paper, we present a method capable of ensuring practical prescribed-time control with guaranteed performance for a class of nonlinear systems in the presence of time-varying parametric and dynamic uncertainties, and uncertain control coefficients. Our design consists of two key steps. First, we construct a performance-rate function that freezes at and after a user-specified time T, playing a crucial role in achieving desired precision within prescribed time T and dealing with unmodeled dynamics. Next, based on this function and a sigma-modification strategy in which the leakage term starts to vanish at t > T, we develop an adaptive dynamic surface control framework to reduce control complexity, deal with uncertainties, ensure prescribed performance, practical prescribed-time convergence to a specific region, and ultimately achieve asymptotic convergence. The effectiveness of the proposed control method is validated through numerical simulations.
Paper Structure (9 sections, 6 theorems, 40 equations, 5 figures, 1 table)

This paper contains 9 sections, 6 theorems, 40 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. System dynamics
  4. Control Objective
  5. Main results
  6. Performance-Rate Function
  7. Controller design
  8. Simulation results
  9. Conclusion

Key Result

Lemma 1

If $V$ is an exp-ISpS Lyapunov function for the system $\dot{\xi}=q(\xi,x,t)$, i.e., the conditions in eq:ISpS1 hold, then, for any constant $\bar{c}\in(0, c)$, initial instant $t_0 > 0$, initial condition $\xi_0 = \xi(t_0)$, continuous function $\bar{\Upsilon}(x_1)$ such that $\bar{\Upsilon}(x_1)\g such that $D(t_0, t) = 0$ for $t \geq t_0 + T_0$, and $V (\xi) \leq r(t) + D(t_0, t)$ with $D(t_0,

Figures (5)

  • Figure 1: Schematic illustration of the time-varying gain $\sigma_2(t)$
  • Figure 2: (a) State trajectory; (b) control input; (c) estimated parameter ($\bar{\sigma}=100$: solid line, $\bar{\sigma}=50$: dash-dotted line, $\bar{\sigma}=30$: dashed line, $\bar{\sigma}=20$: dotted line)
  • Figure 3: (a) Output tracking error ($\rho(t)$: dash-dotted line, $e(t)$ under proposed: solid line, $e(t)$ under zhang2017adaptive: dashed line); (b) First state ($y_d(t)$: dash-dotted line, $x_1(t)$ under proposed: solid line, $x_1(t)$ under zhang2017adaptive: dashed line); (c) Second state ($x_2(t)$ under proposed: solid line, $x_2(t)$ under zhang2017adaptive: dashed line)
  • Figure 4: (a) Control input ($u(t)$ under proposed: solid line, $u(t)$ under zhang2017adaptive: dashed line); (b) dynamic signal ($\xi(t)$: dash-dotted line, $r(t)$ under proposed: solid line, $r(t)$ under zhang2017adaptive: dashed line)
  • Figure 5: (a) Filter error ($\omega_1(t)$ under proposed: solid line, $\omega_1(t)$ under zhang2017adaptive: dashed line); (b) Estimated parameters ($\hat{\gamma}_1(t)$: dash-dotted line, $\hat{\vartheta}_1(t)$: solid line, $\hat{\vartheta}_2(t)$: dashed line); (c) Time-varying gains ($\sigma_1(t)$: solid line, $\sigma_2(t)$: dashed line)

Theorems & Definitions (18)

  • Remark 1
  • Lemma 1: jiang1998design
  • Remark 2
  • Lemma 2: jiang1998design
  • Lemma 3: jiang1998design
  • Lemma 4: li2022command
  • Lemma 5: ren2010adaptive
  • Definition 1
  • Remark 3
  • Remark 4
  • ...and 8 more