# Adaptive Backstepping Design of a Microgyroscope

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Backstepping is a nonlinear control approach based on Lyapunov stability theorem by means of recursion process. Backstepping design is a powerful tool for dynamic systems with pure or strict feedback forms. A major advantage of backstepping is that it has the flexibility to avoid cancellations of useful nonlinearities and achieve regulation and tracking properties. However, the vibratory microgyroscope is neither of these two forms. Therefore, the microgyroscope motion equations should be transformed into a cascade-like system to be suitable for the backstepping approach.
- (2)
- An adaptive control strategy is deployed in the backstepping procedure to deal with parameter uncertainties and external disturbances. The Lyapunov-based adaptive controller is obtained to guarantee the asymptotic stability of the closed-loop system and the consistent parameter estimates, including the external angular velocity if the persistent excitation (PE) condition is satisfied. In addition, a robust term is incorporated in the adaptive backstepping algorithm to suppress the lumped disturbances.

## 2. Microgyroscope Dynamics

## 3. Adaptive Backstepping Control Design

## 4. Adaptive Estimator

**Theorem**

**1.**

**Proof.**

## 5. Simulation Study

_{1}= 6.17, ω

_{2}= 5.11. Here ${\omega}_{1},{\omega}_{2}$ were chosen to be the resonance frequencies of the z-axis MEMS vibratory gyroscope. We assumed that ${\omega}_{1},{\omega}_{2}$ were fixed in the simulation period.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**Tracking errors using the adaptive controller in [2].

**Figure 10.**Adaptation of parameter estimates using the adaptive controller in [2].

**Figure 11.**Control efforts for a microgyroscope using the adaptive controller in [2].

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**MDPI and ACS Style**

Fang, Y.; Fei, J.; Yang, Y.
Adaptive Backstepping Design of a Microgyroscope. *Micromachines* **2018**, *9*, 338.
https://doi.org/10.3390/mi9070338

**AMA Style**

Fang Y, Fei J, Yang Y.
Adaptive Backstepping Design of a Microgyroscope. *Micromachines*. 2018; 9(7):338.
https://doi.org/10.3390/mi9070338

**Chicago/Turabian Style**

Fang, Yunmei, Juntao Fei, and Yuzheng Yang.
2018. "Adaptive Backstepping Design of a Microgyroscope" *Micromachines* 9, no. 7: 338.
https://doi.org/10.3390/mi9070338