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Adaptive backstepping control for FOS with nonsmooth nonlinearities

Dian Sheng, Yiheng Wei, Songsong Cheng, Yong Wang

TL;DR

The paper tackles the problem of controlling incommensurate fractional-order systems with simultaneous dead-zone and input-saturation under uncertainty. It introduces an input-decomposition strategy using an intermediate variable to transform nonlinearities into disturbance and saturation, and applies a fractional-order adaptive backstepping control framework (FOABC) augmented with a frequency-distributed surrogate state and a fractional-order tracking differentiator. Key contributions include transforming the plant into a normalized chain form, developing two FOABC schemes for known and unknown input gains, and providing Lyapunov-based proofs of global boundedness and asymptotic tracking, with corollaries offering flexible nonlinear stabilizers. The approach is validated by simulations demonstrating robust tracking of a sinusoidal reference even under full gain uncertainty and nonsmooth actuator dynamics, highlighting practical applicability for FOS with actuator nonlinearities.

Abstract

This paper proposes an original solution to input saturation and dead zone of fractional order system. To overcome these nonsmooth nonlinearities, the control input is decomposed into two independent parts by introducing an intermediate variable, and thus the problem of dead zone and saturation transforms into the problem of disturbance and saturation afterwards. With the procedure of fractional order adaptive backstepping controller design, the bound of disturbance is estimated, and saturation is compensated by the virtual signal of an auxiliary system as well. In spite of the existence of nonsmooth nonlinearities, the output is guaranteed to track the reference signal asymptotically on the basis of our proposed method. Some simulation studies are carried out in order to demonstrate the effectiveness of method at last.

Adaptive backstepping control for FOS with nonsmooth nonlinearities

TL;DR

The paper tackles the problem of controlling incommensurate fractional-order systems with simultaneous dead-zone and input-saturation under uncertainty. It introduces an input-decomposition strategy using an intermediate variable to transform nonlinearities into disturbance and saturation, and applies a fractional-order adaptive backstepping control framework (FOABC) augmented with a frequency-distributed surrogate state and a fractional-order tracking differentiator. Key contributions include transforming the plant into a normalized chain form, developing two FOABC schemes for known and unknown input gains, and providing Lyapunov-based proofs of global boundedness and asymptotic tracking, with corollaries offering flexible nonlinear stabilizers. The approach is validated by simulations demonstrating robust tracking of a sinusoidal reference even under full gain uncertainty and nonsmooth actuator dynamics, highlighting practical applicability for FOS with actuator nonlinearities.

Abstract

This paper proposes an original solution to input saturation and dead zone of fractional order system. To overcome these nonsmooth nonlinearities, the control input is decomposed into two independent parts by introducing an intermediate variable, and thus the problem of dead zone and saturation transforms into the problem of disturbance and saturation afterwards. With the procedure of fractional order adaptive backstepping controller design, the bound of disturbance is estimated, and saturation is compensated by the virtual signal of an auxiliary system as well. In spite of the existence of nonsmooth nonlinearities, the output is guaranteed to track the reference signal asymptotically on the basis of our proposed method. Some simulation studies are carried out in order to demonstrate the effectiveness of method at last.

Paper Structure

This paper contains 9 sections, 6 theorems, 69 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Model transformation
  5. Input decomposition
  6. FOABC with known $\bar{b}$
  7. FOABC with unknown $\bar{b}$
  8. Simulation Study
  9. Conclusions

Key Result

Lemma 1

(see Montseny:1998LAAS) The differential equation ${{\mathscr D}^{\alpha }}y\left(t \right)=u\left(t \right)$ with fractional order $0<\alpha<1$, ${y\left(t \right) \in \mathbb{R}}$ and ${u\left(t \right) \in \mathbb{R}}$ can be transformed into the following linear continuous frequency distributed where ${{\mu _\alpha }\left( \omega \right) = {{{\omega ^{ - \alpha }}\sin \left( {\alpha \pi } \r

Figures (12)

  • Figure 1: Control input subject to dead zone and saturation.
  • Figure 2: The decomposition of control input.
  • Figure 3: The intermediary $w$ between $v$ and $u$.
  • Figure 4: The performance of tracking in Example 1.
  • Figure 5: The estimation of $\theta$ in Example 1.
  • ...and 7 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • ...and 2 more